Confidence Limits in Statistics

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Questions and Answers

What is the purpose of the industrial hygienist's exposure assessment in the shoe factory?

  • To identify the source of toluene in the factory
  • To estimate the average employee exposure to toluene (correct)
  • To determine the TLV for toluene
  • To measure the air quality in the factory

What is the total sampling time for the 5 employees?

  • 34 hours
  • 30 hours (correct)
  • 32 hours
  • 28 hours

What is the formula to calculate the average concentration?

  • (∑x)/n (correct)
  • ∑x/n
  • n/∑x
  • x/∑n

What is the average concentration of toluene in the sample?

<p>23 ppm (A)</p> Signup and view all the answers

Why is the industrial hygienist concerned about the exposure to toluene?

<p>Because the TLV for toluene is 50 ppm (C)</p> Signup and view all the answers

What is the purpose of calculating the 95% confidence limits for the average employee exposure?

<p>To estimate the range of possible values for the average exposure (A)</p> Signup and view all the answers

What percentage of the area under the normal distribution is represented by the z-value of 1.64?

<p>45% (B)</p> Signup and view all the answers

What is the 5th percentile of the given distribution?

<p>124 (B)</p> Signup and view all the answers

What does a high audit score indicate about a location's safety program?

<p>The location has a good safety program (B)</p> Signup and view all the answers

What does a high injury recordable rate indicate about a location?

<p>The location has a lot of occupational injuries (A)</p> Signup and view all the answers

What is the formula for the standard deviation of a sample?

<p>√(∑(xi - X)^2 / (n - 1)) (C)</p> Signup and view all the answers

What is the correct order of ranking for the audit scores from highest to lowest?

<p>St. Louis, New York, Seattle, Chicago, New Orleans (C)</p> Signup and view all the answers

What is the value of the sample mean in the given data?

<p>23 (D)</p> Signup and view all the answers

What is the Spearman Rank Correlation of safety audit scores and injury recordable rate?

<p>+0.90 (C)</p> Signup and view all the answers

What is the purpose of the z-table in the given problem?

<p>To find the number of standard deviations below the mean (D)</p> Signup and view all the answers

What is the rank of Chicago's audit score?

<p>3 (D)</p> Signup and view all the answers

What is the estimated test score of the 5th percentile?

<p>124 (C)</p> Signup and view all the answers

What is the assumption made about the test scores in the problem?

<p>They are normally distributed (D)</p> Signup and view all the answers

What is the difference (D) in rank between Chicago's audit score and recordable rate?

<p>1 (D)</p> Signup and view all the answers

What is the value of the 95% confidence limits of the average?

<p>8.9, 37.1 (B)</p> Signup and view all the answers

What is the formula for the 95% confidence limits of the average?

<p>X ± 1.96(s / √n) (A)</p> Signup and view all the answers

What is the value of the standard deviation of the test scores?

<p>24 (D)</p> Signup and view all the answers

What is the purpose of calculating the Spearman Rank Correlation?

<p>To measure the correlation between two sets of rankings (D)</p> Signup and view all the answers

What is the formula for calculating the Spearman Rank Correlation (rs)?

<p>rs = 1 - (∑D² / N(N²-1)) (C)</p> Signup and view all the answers

What is the value of N in the formula for calculating the Spearman Rank Correlation?

<p>The number of sites being compared (B)</p> Signup and view all the answers

What is the correct Spearman Rank Correlation for the given data?

<p>-0.9 (C)</p> Signup and view all the answers

What is the purpose of calculating the Spearman Rank Correlation in the given scenario?

<p>To measure the correlation between the Audit Rank and Performance Rank (D)</p> Signup and view all the answers

What is the value of ∑D² in the formula for calculating the Spearman Rank Correlation?

<p>12 + 42 + 42 + 12 + 2² (D)</p> Signup and view all the answers

What is the correct calculation for the Spearman Rank Correlation?

<p>rs = 1 - (∑D² / N(N-1)) (A)</p> Signup and view all the answers

What is the interpretation of a Spearman Rank Correlation of -1?

<p>A perfect negative correlation between the two sets of rankings (D)</p> Signup and view all the answers

What is the formula to calculate Spearman's rank correlation coefficient?

<p>rs = 1 - (6<em>ΣD2) / (N</em>(N-1)) (C)</p> Signup and view all the answers

What is the value of rs in the given dataset?

<p>1/4 (A)</p> Signup and view all the answers

What is the probability distribution used to estimate the probability of a school having four children diagnosed with a CNS cancer in one year?

<p>Poisson distribution (A)</p> Signup and view all the answers

What is the formula to calculate the probability of a Poisson distribution?

<p>P(x) = λx e^(-λ) / x! (D)</p> Signup and view all the answers

What is the estimated probability of a school having four children diagnosed with a CNS cancer in one year?

<p>4.6 x 10^-10 (D)</p> Signup and view all the answers

What is the incidence of CNS cancer in children per million children?

<p>21 per million children (A)</p> Signup and view all the answers

What is the population of the school in the given problem?

<p>500 (A)</p> Signup and view all the answers

What is the purpose of using the Poisson distribution in this problem?

<p>To estimate the probability of a rare event (D)</p> Signup and view all the answers

What is the value of the sample standard deviation in the given data?

<p>16.05 (D)</p> Signup and view all the answers

What is the assumption made about the test scores in the given problem?

<p>The test scores are normally distributed. (A)</p> Signup and view all the answers

What is the value of the average test score for students taking a certain safety exam?

<p>163 (C)</p> Signup and view all the answers

What is the formula to calculate the 95% confidence limits of the average?

<p>X ± 1.96(s/√n) (A)</p> Signup and view all the answers

What is the purpose of the z-table in the given problem?

<p>To determine the number of standard deviations below the mean that provides the 5th percentile. (A)</p> Signup and view all the answers

What is the value of the standard deviation of the test scores?

<p>24 (A)</p> Signup and view all the answers

What is the estimated test score of the 5th percentile?

<p>124 (A)</p> Signup and view all the answers

What is the formula to calculate the sample standard deviation?

<p>s = √(Σ(xi - X)² / (n - 1)) (A)</p> Signup and view all the answers

What is the probability of an employee sustaining an injury due to contact with a sharp object in a particular hour of operation?

<p>1.07 x 10-5 (B)</p> Signup and view all the answers

What is the total number of hours worked by employees in a year?

<p>56,971,309 (C)</p> Signup and view all the answers

What is the average number of injuries per year due to contact with sharp objects?

<p>612 (B)</p> Signup and view all the answers

What is the method used to assign probabilities based on historical data?

<p>Relative frequency method (C)</p> Signup and view all the answers

What is the number of full-time equivalent employees in the corporation?

<p>29,473 (C)</p> Signup and view all the answers

What is the average number of hours worked per employee in a year?

<p>1,933 (A)</p> Signup and view all the answers

What is the purpose of calculating the probability of an injury due to contact with a sharp object?

<p>To assign probabilities based on historical data (A)</p> Signup and view all the answers

What is the synonym for probability?

<p>Likelihood (C)</p> Signup and view all the answers

What is the formula for calculating the Spearman Rank Correlation (rs)?

<p>rs = 1 - 6(∑D²) / (N*(N²-1)) (C)</p> Signup and view all the answers

What is the value of ∑D² in the given dataset?

<p>228 (A)</p> Signup and view all the answers

What is the value of N in the formula for calculating the Spearman Rank Correlation?

<p>4 (A)</p> Signup and view all the answers

What is the interpretation of a Spearman Rank Correlation of -0.9?

<p>A moderate negative correlation between the two variables (D)</p> Signup and view all the answers

What is the value of N in the given dataset?

<p>4 (B)</p> Signup and view all the answers

What is the formula to calculate the probability of a Poisson distribution?

<p>e^(-λ) * (λ^x) / x! (A)</p> Signup and view all the answers

What is the purpose of calculating the Spearman Rank Correlation in the given scenario?

<p>To determine the correlation between two ranked variables (D)</p> Signup and view all the answers

What is the interpretation of a Spearman Rank Correlation of -1?

<p>Perfect negative correlation (A)</p> Signup and view all the answers

What is the correct calculation for the Spearman Rank Correlation?

<p>rs = 1 - (∑D²) / (N*(N²-1)) (A)</p> Signup and view all the answers

What is the value of ∑D² in the formula for calculating the Spearman Rank Correlation?

<p>20 (B)</p> Signup and view all the answers

What is the value of rs in the given dataset?

<p>-0.9 (D)</p> Signup and view all the answers

What is the purpose of using the Poisson distribution in this problem?

<p>To estimate the probability of a school having four children diagnosed with a CNS cancer in one year (B)</p> Signup and view all the answers

What is the rank of the location with the highest audit score?

<p>1 (D)</p> Signup and view all the answers

What is the rank of DC in the given dataset?

<p>4 (C)</p> Signup and view all the answers

What is the value of rs in the given dataset?

<p>-0.5 (D)</p> Signup and view all the answers

What is the estimated probability of a school having four children diagnosed with a CNS cancer in one year?

<p>4.6 x 10^-10 (A)</p> Signup and view all the answers

What is the significance of the z-value of 1.64 in the given problem?

<p>It represents 45% of the area under the normal distribution. (B)</p> Signup and view all the answers

What is the relationship between the audit score and the injury recordable rate?

<p>A high audit score indicates a low injury recordable rate. (A)</p> Signup and view all the answers

What is the purpose of calculating the Spearman Rank Correlation?

<p>To determine the correlation between the audit scores and the injury recordable rates. (A)</p> Signup and view all the answers

What is the rank of St. Louis's audit score?

<p>2 (B)</p> Signup and view all the answers

What is the difference (D) in rank between Chicago's audit score and recordable rate?

<p>1 (C)</p> Signup and view all the answers

What is the interpretation of a Spearman Rank Correlation of +0.90?

<p>There is a strong positive correlation between the audit scores and the injury recordable rates. (D)</p> Signup and view all the answers

What is the correct formula to calculate the Spearman Rank Correlation?

<p>rs = 1 - (6∑D² / N(N²-1)) (D)</p> Signup and view all the answers

What is the value of N in the formula for calculating the Spearman Rank Correlation?

<p>5 (A)</p> Signup and view all the answers

What is the formula for the standard deviation of a sample?

<p>$√(∑ (xi − X)2 / (n-1))$ (A)</p> Signup and view all the answers

What is the assumption made about the test scores in the given problem?

<p>The test scores are normally distributed (A)</p> Signup and view all the answers

What is the purpose of using the z-table in the given problem?

<p>To determine the number of standard deviations below the mean (A)</p> Signup and view all the answers

What is the value of the average test score for students taking a certain safety exam?

<p>163 (A)</p> Signup and view all the answers

What is the value of the standard deviation of the test scores?

<p>24 (A)</p> Signup and view all the answers

What is the formula for the 95% confidence limits of the average?

<p>$X ± 1.96(s / √n)$ (B)</p> Signup and view all the answers

What is the percentage of the area under the normal distribution represented by the z-value?

<p>45% (B)</p> Signup and view all the answers

What is the estimated test score of the 5th percentile?

<p>124 (B)</p> Signup and view all the answers

What is the 5th percentile of the given distribution?

<p>124 (B)</p> Signup and view all the answers

What is the purpose of calculating the confidence limits of the average?

<p>To provide a range of values within which the true mean is likely to lie (D)</p> Signup and view all the answers

What does a high injury recordable rate indicate about a location?

<p>It has a lot of occupational injuries (A)</p> Signup and view all the answers

What is the correct order of ranking for the audit scores from highest to lowest?

<p>St. Louis, New York, Chicago, Seattle, New Orleans (A)</p> Signup and view all the answers

What is the purpose of calculating the Spearman Rank Correlation?

<p>To determine the relationship between safety audit scores and injury recordable rates (A)</p> Signup and view all the answers

What is the correct Spearman Rank Correlation for the given data?

<p>+0.90 (D)</p> Signup and view all the answers

What is the interpretation of a Spearman Rank Correlation of +1?

<p>Perfect positive correlation (D)</p> Signup and view all the answers

What is the purpose of using the z-table in the given problem?

<p>To determine the z-value for a given percentage (C)</p> Signup and view all the answers

What is the formula to calculate the Spearman Rank Correlation?

<p>rs = 1 - (6∑ D2) / (N (N2 - 1)) (D)</p> Signup and view all the answers

What is the purpose of calculating the Spearman Rank Correlation?

<p>To determine the relationship between two ranked variables (B)</p> Signup and view all the answers

What is the value of N in the formula for calculating the Spearman Rank Correlation?

<p>4 (C)</p> Signup and view all the answers

What is the interpretation of a Spearman Rank Correlation of -0.9?

<p>There is a strong negative correlation between the variables (C)</p> Signup and view all the answers

What is the value of ∑D2 in the formula for calculating the Spearman Rank Correlation?

<p>120 (C)</p> Signup and view all the answers

What is the correct Spearman Rank Correlation for the given data?

<p>-0.50 (A)</p> Signup and view all the answers

What is the purpose of the audio visual presentation of the solution?

<p>To provide a visual aid for understanding the solution (B)</p> Signup and view all the answers

What is the copyright year of the material?

<p>2011 (A), 2010 (C)</p> Signup and view all the answers

What is the formula to calculate the Spearman Rank Correlation?

<p>rs = 1 - (6∑D²) / (N * (N - 1)) (D)</p> Signup and view all the answers

What is the probability distribution used to estimate the probability of a school having four children diagnosed with a CNS cancer in one year?

<p>Poisson Distribution (A)</p> Signup and view all the answers

What is the formula for the probability of a Poisson distribution?

<p>P(x) = λx e^(-λ) / x! (B)</p> Signup and view all the answers

What is the population of the school in the given problem?

<p>500 (C)</p> Signup and view all the answers

What is the incidence of CNS cancer in children per million children?

<p>21 per million children (A)</p> Signup and view all the answers

What is the purpose of using the Poisson distribution in this problem?

<p>To estimate the probability of a school having four children diagnosed with a CNS cancer in one year (B)</p> Signup and view all the answers

What is the estimated probability of a school having four children diagnosed with a CNS cancer in one year?

<p>4.6 x 10^-10 (B)</p> Signup and view all the answers

What is the correct calculation for the Spearman Rank Correlation?

<p>rs = 1 - (6∑D²) / (N * (N - 1)) (D)</p> Signup and view all the answers

What is the probability within any one particular hour of operation, that an employee will sustain an injury due to contact with a sharp object?

<p>1.07 x 10-5 (D)</p> Signup and view all the answers

What is the total number of hours worked by employees in a year?

<p>56,971,309 (D)</p> Signup and view all the answers

What is the average number of injuries per year due to contact with sharp objects?

<p>612 (A)</p> Signup and view all the answers

What is the method used to assign probabilities based on historical data?

<p>Relative frequency method (D)</p> Signup and view all the answers

What is the number of full-time equivalent employees who worked in 2011?

<p>29,473 (B)</p> Signup and view all the answers

What is the average number of hours worked by an employee in a year?

<p>1,933 (B)</p> Signup and view all the answers

What is the likelihood of an event occurring?

<p>Probability (A)</p> Signup and view all the answers

What is the unit of time used to calculate the probability of an injury?

<p>Hour (C)</p> Signup and view all the answers

What is the probability of an employee sustaining an injury due to contact with a sharp object in a particular hour of operation?

<p>1.07 x 10^-5 (C)</p> Signup and view all the answers

What percentage of the area under the normal distribution is to the left of the mean?

<p>50% (D)</p> Signup and view all the answers

What is the method used to assign probabilities based on historical data?

<p>Relative frequency method (C)</p> Signup and view all the answers

What is the value of the z-score that corresponds to the 5th percentile of the normal distribution?

<p>-1.64 (A)</p> Signup and view all the answers

What is the purpose of calculating the Spearman Rank Correlation?

<p>To measure the strength and direction of the relationship between two ranked variables (D)</p> Signup and view all the answers

What is the average number of injuries per year due to contact with sharp objects?

<p>612 (D)</p> Signup and view all the answers

What is the total number of hours worked by employees in a year?

<p>56,971,309 (B)</p> Signup and view all the answers

What is the correct interpretation of a Spearman Rank Correlation of +0.90?

<p>There is a strong positive correlation between the two variables (C)</p> Signup and view all the answers

What is the likelihood of an event occurring?

<p>Probability (C)</p> Signup and view all the answers

What is the formula for calculating the Spearman Rank Correlation coefficient?

<p>rs = 1 - (6*∑D² / (n*(n²-1))) (A)</p> Signup and view all the answers

What is the value of the 5th percentile of the distribution given the z-score and the mean and standard deviation?

<p>124 (A)</p> Signup and view all the answers

What is the approach to determine the probability of an event?

<p>Relative frequency method (A)</p> Signup and view all the answers

What is the purpose of using the z-table in the given problem?

<p>To determine the probability of a value occurring by chance (D)</p> Signup and view all the answers

What is the unit of time used to determine the probability of an employee sustaining an injury?

<p>Hour (D)</p> Signup and view all the answers

What is the formula used to calculate the probability of an event?

<p>P(event) = Number of injuries / Total hours worked (C)</p> Signup and view all the answers

What is the correct interpretation of a Spearman Rank Correlation of -1?

<p>There is a strong negative correlation between the two variables (C)</p> Signup and view all the answers

What is the purpose of calculating Spearman Rank Correlation in the given scenario?

<p>To identify the relationship between audit scores and injury recordable rate. (B)</p> Signup and view all the answers

What is the formula for calculating the Spearman Rank Correlation (rs)?

<p>rs = 1 - (6∑D²) / (N (N - 1)) (C)</p> Signup and view all the answers

What is the value of N in the formula for calculating the Spearman Rank Correlation?

<p>4 (D)</p> Signup and view all the answers

What is the interpretation of a Spearman Rank Correlation of -1?

<p>A perfect negative correlation between variables. (C)</p> Signup and view all the answers

What is the probability distribution used to estimate the probability of a school having four children diagnosed with a CNS cancer in one year?

<p>Poisson Distribution (C)</p> Signup and view all the answers

What is the formula to calculate the probability of a Poisson distribution?

<p>P(x) = (λ^x) * e^(-λ) / x! (D)</p> Signup and view all the answers

What is the estimated probability of a school having four children diagnosed with a CNS cancer in one year?

<p>4.6 x 10^-10 (D)</p> Signup and view all the answers

What is the incidence of CNS cancer in children per million children?

<p>21 per million children (B)</p> Signup and view all the answers

What is the significance of the z-value in a normal distribution?

<p>It represents the standard deviation from the mean. (D)</p> Signup and view all the answers

What is the purpose of calculating the Spearman Rank Correlation?

<p>To identify the correlation between two continuous variables. (A)</p> Signup and view all the answers

What is the interpretation of a Spearman Rank Correlation of +0.90?

<p>There is a strong positive correlation between the two variables. (B)</p> Signup and view all the answers

What is the 5th percentile of a normal distribution?

<p>The lowest 5% of the data points. (B)</p> Signup and view all the answers

What is the assumption made about the data when using the Spearman Rank Correlation?

<p>The data must be continuous. (A)</p> Signup and view all the answers

What is the formula for calculating the Spearman Rank Correlation?

<p>ρ = 1 - (6∑D² / (n(n² - 1))) (B)</p> Signup and view all the answers

What is the purpose of using percentiles in statistical analysis?

<p>To identify the outliers in a dataset. (C)</p> Signup and view all the answers

What is the relationship between the z-value and the percentile in a normal distribution?

<p>The z-value is the standard deviation from the mean, and the percentile is the proportion of data points below the z-value. (B)</p> Signup and view all the answers

What is the definition of likelihood in statistics?

<p>A synonym for probability (C)</p> Signup and view all the answers

What is the purpose of the relative frequency method in statistical analysis?

<p>To determine the probability of an event based on historical data (B)</p> Signup and view all the answers

What is the probability of an employee sustaining an injury due to contact with a sharp object in a particular hour of operation?

<p>1.07 x 10^-5 (C)</p> Signup and view all the answers

What is the total number of hours worked by employees in a year?

<p>56,971,309 (A)</p> Signup and view all the answers

What is the average number of injuries per year due to contact with sharp objects?

<p>612 (D)</p> Signup and view all the answers

What is the formula to calculate the probability of an event using the Poisson distribution?

<p>e^(-λ) * (λ^k) / k! (B)</p> Signup and view all the answers

What is the difference between a probability and a rate?

<p>A probability is a measure of the likelihood of an event, while a rate is a measure of the frequency of an event (B)</p> Signup and view all the answers

What is the main assumption of the Poisson distribution?

<p>The events are independent and identically distributed (D)</p> Signup and view all the answers

What is the interpretation of a Spearman Rank Correlation of -1 in the context of safety audit scores and injury recordable rate?

<p>A perfect negative correlation between safety audit scores and injury recordable rate (B)</p> Signup and view all the answers

What is the purpose of calculating the 95% confidence limits of the average employee exposure?

<p>To estimate the range of values within which the true population mean is likely to lie (B)</p> Signup and view all the answers

What is the assumption made about the test scores in the given problem?

<p>The test scores follow a normal distribution (B)</p> Signup and view all the answers

What is the formula for calculating the probability of a Poisson distribution?

<p>P(x) = (e^(-) * (^x)) / x! (B)</p> Signup and view all the answers

What is the value of N in the formula for calculating the Spearman Rank Correlation?

<p>The number of data points (B)</p> Signup and view all the answers

What is the purpose of using the Poisson distribution in this problem?

<p>To model the number of children diagnosed with CNS cancer in one year (C)</p> Signup and view all the answers

What is the interpretation of a Spearman Rank Correlation of 0.5 in the context of safety audit scores and injury recordable rate?

<p>A moderate positive correlation between safety audit scores and injury recordable rate (A)</p> Signup and view all the answers

What is the formula for calculating the Spearman Rank Correlation?

<p>rs = 1 - (6 * D) / (N * (N - 1)) (A), rs = 1 - (6 * D) / (N * (N - 1)) (B)</p> Signup and view all the answers

What is the z-value corresponding to 45% of the area under the normal distribution?

<p>1.64 (B)</p> Signup and view all the answers

What is the 5th percentile of the given distribution?

<p>124 (C)</p> Signup and view all the answers

What is the purpose of calculating the Spearman Rank Correlation?

<p>To measure the strength of association between two variables (C)</p> Signup and view all the answers

What is the rank of Chicago's audit score?

<p>3 (B)</p> Signup and view all the answers

What is the formula for calculating the Spearman Rank Correlation?

<p>rs = 1 - (6∑D²)/(n(n²-1)) (A)</p> Signup and view all the answers

What is the interpretation of a Spearman Rank Correlation of +0.90?

<p>There is a strong positive correlation between the two variables (B)</p> Signup and view all the answers

What is the purpose of using the z-table in the given problem?

<p>To find the z-value corresponding to a given percentile (B)</p> Signup and view all the answers

What is the value of N in the formula for calculating the Spearman Rank Correlation?

<p>5 (A)</p> Signup and view all the answers

What is the probability of an employee sustaining an injury due to contact with a sharp object in a particular hour of operation?

<p>1.07 x 10^-5 (C)</p> Signup and view all the answers

What is the total number of hours worked by employees in a year?

<p>56,971,309 (B)</p> Signup and view all the answers

What is the average number of injuries per year due to contact with sharp objects?

<p>612 (B)</p> Signup and view all the answers

What is the method used to assign probabilities based on historical data?

<p>Relative frequency method (A)</p> Signup and view all the answers

What is the meaning of the term 'likelihood' in the context of statistics?

<p>Probability of an event (C)</p> Signup and view all the answers

What is the purpose of calculating the probability of an injury due to contact with a sharp object?

<p>To assign a probability based on historical data (D)</p> Signup and view all the answers

What is the unit of time for which the probability of an injury due to contact with a sharp object is calculated?

<p>Hour (B)</p> Signup and view all the answers

What is the reason for calculating the probability of an injury due to contact with a sharp object?

<p>To assign a probability based on historical data (D)</p> Signup and view all the answers

What is the interpretation of a Spearman Rank Correlation of -1?

<p>There is a perfect negative correlation between the two variables. (A)</p> Signup and view all the answers

What is the purpose of calculating the 95% confidence limits of the average?

<p>To estimate the range of values within which the population mean is likely to lie. (C)</p> Signup and view all the answers

What is the formula to calculate the probability of a Poisson distribution?

<p>P(x) = λx e^(-λ) / x! (C)</p> Signup and view all the answers

What is the value of N in the formula for calculating the Spearman Rank Correlation?

<p>The number of observations in the sample. (D)</p> Signup and view all the answers

What is the assumption made about the test scores in the given problem?

<p>The test scores are normally distributed. (D)</p> Signup and view all the answers

What is the purpose of using the Poisson distribution in this problem?

<p>To estimate the probability of a discrete outcome. (C)</p> Signup and view all the answers

What is the formula for the standard deviation of a sample?

<p>∑ (xi - X)² / (n - 1) (C)</p> Signup and view all the answers

What is the purpose of calculating the 95% confidence limits of the average?

<p>To estimate the population mean (A)</p> Signup and view all the answers

What is the assumption made about the test scores in the problem?

<p>They are normally distributed (D)</p> Signup and view all the answers

What is the value of the standard deviation of the test scores?

<p>24 (C)</p> Signup and view all the answers

What is the estimated test score of the 5th percentile?

<p>124 (A)</p> Signup and view all the answers

What is the formula for the 95% confidence limits of the average?

<p>X ± 1.96(s / √n) (C)</p> Signup and view all the answers

What percentage of the area under the normal distribution is left after subtracting 50% and 5%?

<p>45% (B)</p> Signup and view all the answers

What is the purpose of the z-table in the given problem?

<p>To find the percentile of a given score (C)</p> Signup and view all the answers

What is the mean minus 1.64 standard deviations equal to?

<p>124 (A)</p> Signup and view all the answers

What is the value of the average test score for students taking a certain safety exam?

<p>163 (C)</p> Signup and view all the answers

What does a high audit score indicate about a location?

<p>It has a good safety program (A)</p> Signup and view all the answers

What is the correct direction of the Spearman Rank Correlation between audit scores and injury recordable rates?

<p>Positive (A)</p> Signup and view all the answers

What is the purpose of calculating the Spearman Rank Correlation?

<p>To identify the relationship between two variables (A)</p> Signup and view all the answers

What is the rank of Chicago's audit score in the given data?

<p>3 (C)</p> Signup and view all the answers

How many locations are included in the given data for calculating the Spearman Rank Correlation?

<p>5 (B)</p> Signup and view all the answers

What is the correct calculation for the Spearman Rank Correlation?

<p>1 - (6 * ∑D²) / (N * (N² - 1)) (B)</p> Signup and view all the answers

What is the formula for the Spearman Rank Correlation coefficient (rs)?

<p>rs = 1 - (6∑D²)/(N*(N-1)) (B)</p> Signup and view all the answers

What is the probability distribution used to estimate the probability of a school having four children diagnosed with a CNS cancer in one year?

<p>Poisson Distribution (A)</p> Signup and view all the answers

What is the estimated probability of a school having four children diagnosed with a CNS cancer in one year?

<p>4.6 x 10^-10 (D)</p> Signup and view all the answers

What is the incidence of CNS cancer in children per million children?

<p>21 per million children (A)</p> Signup and view all the answers

What is the population of the school in the given problem?

<p>500 (D)</p> Signup and view all the answers

What is the formula for the Poisson Distribution?

<p>P(x) = (e^(-λ) * (λx) / x!) (A)</p> Signup and view all the answers

What is the purpose of using the Poisson Distribution in this problem?

<p>To estimate the probability of a school having four children diagnosed with a CNS cancer in one year (A)</p> Signup and view all the answers

What is the formula for calculating the Spearman Rank Correlation?

<p>rs = 1 - (6∑ D2 / (N(N2 - 1))) (D)</p> Signup and view all the answers

What is the value of N in the formula for calculating the Spearman Rank Correlation?

<p>4 (D)</p> Signup and view all the answers

What is the value of N in the formula for calculating the Spearman Rank Correlation?

<p>4 (B)</p> Signup and view all the answers

What is the interpretation of a Spearman Rank Correlation of -0.9?

<p>There is a strong negative correlation between the two variables. (B)</p> Signup and view all the answers

What is the purpose of calculating the Spearman Rank Correlation?

<p>To determine the rank correlation between two variables. (C)</p> Signup and view all the answers

What is the correct Spearman Rank Correlation for the given data in the problem?

<p>-0.90 (C)</p> Signup and view all the answers

What is the value of ∑D² in the formula for calculating the Spearman Rank Correlation?

<p>120 (B)</p> Signup and view all the answers

What is the correct calculation for the Spearman Rank Correlation?

<p>rs = 1 - (6∑ D2 / (N(N2 - 1))) (D)</p> Signup and view all the answers

What is the purpose of the Spearman Rank Correlation in the given scenario?

<p>To determine the rank correlation between the audit rank and performance rank. (A)</p> Signup and view all the answers

What is the purpose of determining the probability of an employee sustaining an injury due to contact with a sharp object?

<p>To assign probabilities based on historical data (A)</p> Signup and view all the answers

What is the average number of injuries per year due to contact with sharp objects?

<p>612 (A)</p> Signup and view all the answers

What is the total number of hours worked by employees in a year?

<p>56,971,309 (A)</p> Signup and view all the answers

What is the probability of an employee sustaining an injury due to contact with a sharp object in a particular hour of operation?

<p>1.07 x 10-5 (B)</p> Signup and view all the answers

What is likelihood a synonym for?

<p>Probability (D)</p> Signup and view all the answers

What is the method used to assign probabilities based on historical data?

<p>Relative frequency method (D)</p> Signup and view all the answers

What is the number of full-time equivalent employees who worked in a year?

<p>29,473 (B)</p> Signup and view all the answers

What is the average number of hours worked by an employee in a year?

<p>1,933 (A)</p> Signup and view all the answers

What is the z-value corresponding to 45% of the area under the normal distribution?

<p>1.64 (C)</p> Signup and view all the answers

What is the formula for calculating the Spearman Rank Correlation?

<p>rs = 1 - 6∑ D² / (N(N²-1)) (A)</p> Signup and view all the answers

What is the 5th percentile of the given distribution?

<p>124 (B)</p> Signup and view all the answers

What is the value of N in the formula for calculating the Spearman Rank Correlation?

<p>4 (D)</p> Signup and view all the answers

What is the purpose of calculating the Spearman Rank Correlation?

<p>To identify the relationship between audit scores and injury recordable rates (A)</p> Signup and view all the answers

What is the interpretation of a Spearman Rank Correlation of -1?

<p>A perfect negative correlation (C)</p> Signup and view all the answers

What is the rank of Chicago's audit score in the given data?

<p>3 (C)</p> Signup and view all the answers

What is the value of ∑D² in the formula for calculating the Spearman Rank Correlation?

<p>228 (D)</p> Signup and view all the answers

What is the formula for calculating the Spearman Rank Correlation (rs)?

<p>rs = 1 - (6∑D²) / (n(n²-1)) (A)</p> Signup and view all the answers

What is the formula for the standard deviation of a sample?

<p>$$s = \sqrt{\sum{(x_i - X)^2 / (n-1)}}$$ (A)</p> Signup and view all the answers

What is the interpretation of a Spearman Rank Correlation of +0.90?

<p>There is a strong positive correlation between the variables (D)</p> Signup and view all the answers

What is the purpose of calculating the Spearman Rank Correlation?

<p>To measure the correlation between two ranked variables (D)</p> Signup and view all the answers

What is the value of the average test score for students taking a certain safety exam?

<p>163 (C)</p> Signup and view all the answers

What is the correct calculation for the Spearman Rank Correlation?

<p>rs = 1 - 6∑ D² / (N(N²-1)) (B)</p> Signup and view all the answers

What is the difference (D) in rank between Chicago's audit score and recordable rate?

<p>1 (A)</p> Signup and view all the answers

What is the correct Spearman Rank Correlation for the given data?

<p>+0.90 (D)</p> Signup and view all the answers

What is the value of rs in the given dataset?

<p>-0.9 (C)</p> Signup and view all the answers

What is the purpose of using the z-table in the given problem?

<p>To determine the number of standard deviations below the mean that provides the 5th percentile (D)</p> Signup and view all the answers

What is the value of the standard deviation of the test scores?

<p>24 (D)</p> Signup and view all the answers

What is the rank of the site with the highest audit score?

<p>1 (A)</p> Signup and view all the answers

What is the estimated test score of the 5th percentile?

<p>124 (D)</p> Signup and view all the answers

What is the formula for the 95% confidence limits of the average?

<p>$$X \pm 1.96 * (s / \sqrt{n})$$ (B)</p> Signup and view all the answers

What assumption is made about the test scores in the given problem?

<p>The test scores are normally distributed (B)</p> Signup and view all the answers

What is the purpose of calculating the 95% confidence limits of the average?

<p>To provide a range of values within which the true population mean is likely to lie (D)</p> Signup and view all the answers

What is the formula to calculate the Spearman Rank Correlation (rs)?

<p>rs = 1 − (6∑D²) / (N (N - 1)) (A)</p> Signup and view all the answers

What is the value of ∑D² in the given dataset?

<p>42 (A)</p> Signup and view all the answers

What is the probability distribution used to estimate the probability of a school having four children diagnosed with a CNS cancer in one year?

<p>Poisson distribution (A)</p> Signup and view all the answers

What is the formula to calculate the probability of a Poisson distribution?

<p>P(x) = (e^(-λ) × (λ^x)) / x! (B)</p> Signup and view all the answers

What is the incidence of CNS cancer in children per million children?

<p>21 per million children (A)</p> Signup and view all the answers

What is the correct calculation for the Spearman Rank Correlation?

<p>rs = 1 - (6∑D²) / (N (N - 1)) (B)</p> Signup and view all the answers

What is the population of the school in the given problem?

<p>500 (A)</p> Signup and view all the answers

What is the estimated probability of a school having four children diagnosed with a CNS cancer in one year?

<p>4.6 x 10^-10 (D)</p> Signup and view all the answers

What is the probability of an employee sustaining an injury due to contact with a sharp object in a particular hour of operation?

<p>1.07 x 10^-5 (C)</p> Signup and view all the answers

What is the total number of hours worked by employees in a year?

<p>56,971,309 (B)</p> Signup and view all the answers

What is the average number of injuries per year due to contact with sharp objects?

<p>612 (B)</p> Signup and view all the answers

What is the method used to assign probabilities based on historical data?

<p>Relative frequency method (A)</p> Signup and view all the answers

What is the formula to calculate the probability of an injury due to contact with a sharp object?

<p>number of injuries / total hours worked (D)</p> Signup and view all the answers

What is the likelihood of an event occurring?

<p>Probability (B)</p> Signup and view all the answers

What is the purpose of calculating the probability of an injury due to contact with a sharp object?

<p>To assign probabilities based on historical data (C)</p> Signup and view all the answers

What is the unit of time used to determine the probability of an injury due to contact with a sharp object?

<p>Hour (B)</p> Signup and view all the answers

What is the correct variance of the sample concentrations?

<p>45.2 (A)</p> Signup and view all the answers

What is the correct standard deviation of the sample concentrations?

<p>6.71 (D)</p> Signup and view all the answers

What is the correct margin of error for the 95% confidence interval?

<p>14.1 (B)</p> Signup and view all the answers

What is the correct t-score for the given sample and confidence level?

<p>1.86 (B)</p> Signup and view all the answers

What is the correct degrees of freedom for the given sample?

<p>4 (D)</p> Signup and view all the answers

What is the correct critical value for the 95% confidence interval?

<p>2.13 (B)</p> Signup and view all the answers

What is the correct calculation for the sample mean?

<p>∑x / n (C)</p> Signup and view all the answers

What is the correct interpretation of the 95% confidence interval?

<p>There is a 95% chance that the true mean is within the interval. (D)</p> Signup and view all the answers

What is the formula for calculating the standard deviation of a sample?

<p>∑(xi - X)^2 / (n - 1) (C)</p> Signup and view all the answers

What is the purpose of calculating the 95% confidence limits of the average?

<p>To determine the range of possible values for the population mean (C)</p> Signup and view all the answers

What is the assumption made about the test scores in the given problem?

<p>The test scores are normally distributed (A)</p> Signup and view all the answers

What is the value of the z-score corresponding to the 5th percentile?

<p>-1.64 (D)</p> Signup and view all the answers

What is the formula for calculating the 95% confidence limits of the average?

<p>X ± (1.96 * s / sqrt(n)) (A)</p> Signup and view all the answers

What is the value of the sample standard deviation in the given data?

<p>16.05 (B)</p> Signup and view all the answers

What is the estimated test score of the 5th percentile?

<p>124 (C)</p> Signup and view all the answers

What is the purpose of the z-table in the given problem?

<p>To determine the z-score corresponding to a certain percentage (A)</p> Signup and view all the answers

What is the purpose of calculating the Spearman Rank Correlation in the given scenario?

<p>To identify the correlation between the audit scores and injury recordable rates (C)</p> Signup and view all the answers

What is the formula to calculate the probability of a Poisson distribution?

<p>P(x) = λ x e-λ / x! (B)</p> Signup and view all the answers

What is the interpretation of a Spearman Rank Correlation of -1?

<p>A perfect negative correlation between the two variables (B)</p> Signup and view all the answers

What is the formula to calculate the Spearman Rank Correlation coefficient (rs)?

<p>rs = 1 - 6∑ D² / (N(N² - 1)) (D)</p> Signup and view all the answers

What is the value of N in the formula for calculating the Spearman Rank Correlation?

<p>The total number of observations in the dataset (A)</p> Signup and view all the answers

What is the estimated probability of a school having four children diagnosed with a CNS cancer in one year?

<p>4.6 x 10^-10 (A)</p> Signup and view all the answers

What is the interpretation of a Spearman Rank Correlation of -1?

<p>A perfect negative correlation between the two ranked variables (A)</p> Signup and view all the answers

What is the value of N in the formula for calculating the Spearman Rank Correlation?

<p>The number of observations in the dataset (D)</p> Signup and view all the answers

What is the value of ∑D² in the formula for calculating the Spearman Rank Correlation?

<p>20 (A)</p> Signup and view all the answers

What is the purpose of calculating the Spearman Rank Correlation in the given scenario?

<p>To determine the correlation between the audit scores and performance ranks (C)</p> Signup and view all the answers

What is the formula for calculating the Spearman Rank Correlation?

<p>rs = 1 - (6∑D² / N(N-1)) (B)</p> Signup and view all the answers

What is the assumption made about the data in the Poisson distribution?

<p>The data follows a Poisson distribution, which is a discrete distribution used to model the number of occurrences of a rare event (A)</p> Signup and view all the answers

What is the value of ∑D² in the formula for calculating the Spearman Rank Correlation?

<p>12 + 4² + 4² + 12 + 2² (C)</p> Signup and view all the answers

What is the correct calculation for the Spearman Rank Correlation?

<p>rs = 1 - 6(12 + 4² + 4² + 12 + 2²) / (5(5² - 1)) (D)</p> Signup and view all the answers

What is the value of rs in the given dataset?

<p>-0.90 (A)</p> Signup and view all the answers

What is the interpretation of the Spearman Rank Correlation coefficient in the given scenario?

<p>There is a strong negative correlation between the audit scores and performance ranks (D)</p> Signup and view all the answers

What is the probability of an employee sustaining an injury due to contact with a sharp object in a particular hour of operation?

<p>1.07 x 10-5 (A)</p> Signup and view all the answers

What is the method used to assign probabilities based on historical data?

<p>Relative frequency method (A)</p> Signup and view all the answers

What is the average number of injuries per year due to contact with sharp objects?

<p>612 (B)</p> Signup and view all the answers

What is the total number of hours worked by employees in a year?

<p>56,971,309 (B)</p> Signup and view all the answers

What is the likelihood of an employee sustaining an injury due to contact with a sharp object in a particular hour of operation?

<p>1.07 x 10-5 (D)</p> Signup and view all the answers

What is the probability of an employee sustaining an injury due to contact with a sharp object in a year?

<p>0.0105 (B)</p> Signup and view all the answers

What is the formula to calculate the probability of an injury due to contact with a sharp object?

<p>P(injury) = (sharp object injuries / total hours worked) (C)</p> Signup and view all the answers

What is the significance of the 'relative frequency method' in assigning probabilities?

<p>It is used to assign probabilities based on historical data. (A)</p> Signup and view all the answers

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Study Notes

Confidence Limits

  • The average concentration of toluene in a shoe factory is calculated by summing the concentrations and dividing by the number of samples.
  • The standard deviation is calculated using the formula: s = sqrt((Σ(xi - X)^2) / (n - 1)), where xi is the concentration of each sample, X is the average concentration, and n is the number of samples.
  • The 95% confidence limits are calculated using the formula: X ± 1.96 * (s / sqrt(n)), where X is the average concentration, s is the standard deviation, and n is the number of samples.

Z-Table

  • The z-table is used to determine the number of standard deviations below the mean that corresponds to a certain percentile.
  • The z-value is calculated using the formula: z = (X - μ) / σ, where X is the value, μ is the mean, and σ is the standard deviation.
  • The 5th percentile is calculated by subtracting 1.64 standard deviations from the mean, where 1.64 is the z-value corresponding to 45% of the area under the normal distribution.

Spearman Rank Correlation

  • The Spearman Rank Correlation is calculated using the formula: rs = 1 - (6 * ΣD^2) / (n * (n^2 - 1)), where D is the difference in rank between two variables, and n is the number of samples.
  • The correlation coefficient ranges from -1 to 1, where -1 indicates perfect negative correlation, 0 indicates no correlation, and 1 indicates perfect positive correlation.

Poisson Distribution

  • The Poisson distribution is used to estimate the probability of a certain number of events occurring in a fixed interval of time or space, where the events occur with a known average rate.
  • The probability of a certain number of events occurring is calculated using the formula: P(x) = (λ^x * e^(-λ)) / x!, where λ is the average rate of events, and x is the number of events.
  • The probability of four children being diagnosed with CNS cancer in one year is estimated using the Poisson distribution, where the average rate of CNS cancer is 21 per million children.

Statistics and Probability

  • The standard deviation is given by the formula: s = sqrt((∑(xi - X)^2) / (n - 1))
  • The 95% confidence limits of the average are given by: X ± 1.96(s / sqrt(n))

Z-Table and Normal Distribution

  • The z-table is used to determine the number of standard deviations below the mean that provides the 5th percentile
  • The z-value with 45% of the area under the normal distribution is 1.64
  • The 5th percentile is given by the mean subtracting 1.64 standard deviations: 163 - 1.64(24) = 124

Spearman Rank Correlation

  • The formula for Spearman Rank Correlation is: rs = 1 - (6∑D^2) / (N(N^2 - 1))
  • The difference (D) in rank between the two scores is calculated: D = Audit Rank - Recordable Rank
  • Example: rs = 1 - (6(1^2 + 4^2 + 4^2 + 1^2 + 2^2)) / (5(5^2 - 1)) = -0.9

Poisson Distribution

  • The formula for Poisson Distribution is: P(x) = (λ^x * e^(-λ)) / x!
  • The expected frequency λ is calculated: λ = population x frequency
  • Example: λ = 500 x (21/1,000,000) = 0.0105
  • The probability of a school having four children diagnosed with a CNS cancer in one year is: P(4) = (0.0105^4 * e^(-0.0105)) / 4! = 4.6 x 10^-10

Probability of Injury

  • The probability of an injury due to contact with a sharp object is calculated: P(injury) = 612 sharp object injuries / 56,971,309 hours worked
  • The result is: P(injury) = 1.07 x 10^-5
  • The relative frequency method is used to assign probabilities based on historical data.

Statistics and Probability

  • The standard deviation is given by the formula: s = sqrt((∑(xi - X)^2) / (n - 1))
  • The 95% confidence limits of the average are given by: X ± 1.96(s / sqrt(n))

Z-Table and Normal Distribution

  • The z-table is used to determine the number of standard deviations below the mean that provides the 5th percentile
  • The z-value with 45% of the area under the normal distribution is 1.64
  • The 5th percentile is given by the mean subtracting 1.64 standard deviations: 163 - 1.64(24) = 124

Spearman Rank Correlation

  • The formula for Spearman Rank Correlation is: rs = 1 - (6∑D^2) / (N(N^2 - 1))
  • The difference (D) in rank between the two scores is calculated: D = Audit Rank - Recordable Rank
  • Example: rs = 1 - (6(1^2 + 4^2 + 4^2 + 1^2 + 2^2)) / (5(5^2 - 1)) = -0.9

Poisson Distribution

  • The formula for Poisson Distribution is: P(x) = (λ^x * e^(-λ)) / x!
  • The expected frequency λ is calculated: λ = population x frequency
  • Example: λ = 500 x (21/1,000,000) = 0.0105
  • The probability of a school having four children diagnosed with a CNS cancer in one year is: P(4) = (0.0105^4 * e^(-0.0105)) / 4! = 4.6 x 10^-10

Probability of Injury

  • The probability of an injury due to contact with a sharp object is calculated: P(injury) = 612 sharp object injuries / 56,971,309 hours worked
  • The result is: P(injury) = 1.07 x 10^-5
  • The relative frequency method is used to assign probabilities based on historical data.

Statistics and Probability

  • The z-value with 45% of the area under the normal distribution is 1.64.
  • The 5th percentile is calculated by subtracting 1.64 standard deviations from the mean.

Spearman Rank Correlation

  • To calculate the Spearman Rank Correlation, set up a table with rank of audit score and recordable rate.
  • Add an additional column to note the difference (D) in rank between the two scores.
  • Calculate the correlation coefficient (rs) using the formula: rs = 1 - (6∑ D2) / (N (N - 1)).

Poisson Distribution

  • The Poisson distribution is used to estimate the probability of a specific number of events occurring in a fixed interval.
  • The probability of a school having four children diagnosed with a CNS cancer in one year can be estimated using the Poisson distribution.
  • The formula for the Poisson distribution is: P(x) = (λ x e-λ) / x!

Probability

  • Likelihood is a synonym for probability.
  • The relative frequency method is used to assign probabilities based on historical data.
  • The probability of an event occurring can be calculated using the formula: P(event) = number of occurrences / total number of trials.

Specific Problems

  • The probability of an employee sustaining an injury due to contact with a sharp object in a given hour of operation is 1.07 x 10^-5.
  • The Spearman Rank Correlation of safety audit scores and injury recordable rate is +0.90.
  • The probability of a school having four children diagnosed with a CNS cancer in one year is 4.6 x 10^-10.

Statistics and Probability

  • The z-value with 45% of the area under the normal distribution is 1.64.
  • The 5th percentile is calculated by subtracting 1.64 standard deviations from the mean.

Spearman Rank Correlation

  • To calculate the Spearman Rank Correlation, set up a table with rank of audit score and recordable rate.
  • Add an additional column to note the difference (D) in rank between the two scores.
  • Calculate the correlation coefficient (rs) using the formula: rs = 1 - (6∑ D2) / (N (N - 1)).

Poisson Distribution

  • The Poisson distribution is used to estimate the probability of a specific number of events occurring in a fixed interval.
  • The probability of a school having four children diagnosed with a CNS cancer in one year can be estimated using the Poisson distribution.
  • The formula for the Poisson distribution is: P(x) = (λ x e-λ) / x!

Probability

  • Likelihood is a synonym for probability.
  • The relative frequency method is used to assign probabilities based on historical data.
  • The probability of an event occurring can be calculated using the formula: P(event) = number of occurrences / total number of trials.

Specific Problems

  • The probability of an employee sustaining an injury due to contact with a sharp object in a given hour of operation is 1.07 x 10^-5.
  • The Spearman Rank Correlation of safety audit scores and injury recordable rate is +0.90.
  • The probability of a school having four children diagnosed with a CNS cancer in one year is 4.6 x 10^-10.

Statistics and Probability

  • The z-value with 45% of the area under the normal distribution is 1.64.
  • The 5th percentile is calculated by subtracting 1.64 standard deviations from the mean.

Spearman Rank Correlation

  • To calculate the Spearman Rank Correlation, set up a table with rank of audit score and recordable rate.
  • Add an additional column to note the difference (D) in rank between the two scores.
  • Calculate the correlation coefficient (rs) using the formula: rs = 1 - (6∑ D2) / (N (N - 1)).

Poisson Distribution

  • The Poisson distribution is used to estimate the probability of a specific number of events occurring in a fixed interval.
  • The probability of a school having four children diagnosed with a CNS cancer in one year can be estimated using the Poisson distribution.
  • The formula for the Poisson distribution is: P(x) = (λ x e-λ) / x!

Probability

  • Likelihood is a synonym for probability.
  • The relative frequency method is used to assign probabilities based on historical data.
  • The probability of an event occurring can be calculated using the formula: P(event) = number of occurrences / total number of trials.

Specific Problems

  • The probability of an employee sustaining an injury due to contact with a sharp object in a given hour of operation is 1.07 x 10^-5.
  • The Spearman Rank Correlation of safety audit scores and injury recordable rate is +0.90.
  • The probability of a school having four children diagnosed with a CNS cancer in one year is 4.6 x 10^-10.

Statistics and Probability

  • The standard deviation is given by the formula: s = sqrt((∑(xi - X)^2) / (n - 1))
  • The 95% confidence limits of the average are given by: X ± 1.96(s / sqrt(n))

Z-Table and Normal Distribution

  • The z-table is used to determine the number of standard deviations below the mean that provides the 5th percentile
  • The z-value with 45% of the area under the normal distribution is 1.64
  • The 5th percentile is given by the mean subtracting 1.64 standard deviations: 163 - 1.64(24) = 124

Spearman Rank Correlation

  • The formula for Spearman Rank Correlation is: rs = 1 - (6∑D^2) / (N(N^2 - 1))
  • The difference (D) in rank between the two scores is calculated: D = Audit Rank - Recordable Rank
  • Example: rs = 1 - (6(1^2 + 4^2 + 4^2 + 1^2 + 2^2)) / (5(5^2 - 1)) = -0.9

Poisson Distribution

  • The formula for Poisson Distribution is: P(x) = (λ^x * e^(-λ)) / x!
  • The expected frequency λ is calculated: λ = population x frequency
  • Example: λ = 500 x (21/1,000,000) = 0.0105
  • The probability of a school having four children diagnosed with a CNS cancer in one year is: P(4) = (0.0105^4 * e^(-0.0105)) / 4! = 4.6 x 10^-10

Probability of Injury

  • The probability of an injury due to contact with a sharp object is calculated: P(injury) = 612 sharp object injuries / 56,971,309 hours worked
  • The result is: P(injury) = 1.07 x 10^-5
  • The relative frequency method is used to assign probabilities based on historical data.

Statistics and Probability

  • The standard deviation is given by the formula: s = sqrt((∑(xi - X)^2) / (n - 1))
  • The 95% confidence limits of the average are given by: X ± 1.96(s / sqrt(n))

Z-Table and Normal Distribution

  • The z-table is used to determine the number of standard deviations below the mean that provides the 5th percentile
  • The z-value with 45% of the area under the normal distribution is 1.64
  • The 5th percentile is given by the mean subtracting 1.64 standard deviations: 163 - 1.64(24) = 124

Spearman Rank Correlation

  • The formula for Spearman Rank Correlation is: rs = 1 - (6∑D^2) / (N(N^2 - 1))
  • The difference (D) in rank between the two scores is calculated: D = Audit Rank - Recordable Rank
  • Example: rs = 1 - (6(1^2 + 4^2 + 4^2 + 1^2 + 2^2)) / (5(5^2 - 1)) = -0.9

Poisson Distribution

  • The formula for Poisson Distribution is: P(x) = (λ^x * e^(-λ)) / x!
  • The expected frequency λ is calculated: λ = population x frequency
  • Example: λ = 500 x (21/1,000,000) = 0.0105
  • The probability of a school having four children diagnosed with a CNS cancer in one year is: P(4) = (0.0105^4 * e^(-0.0105)) / 4! = 4.6 x 10^-10

Probability of Injury

  • The probability of an injury due to contact with a sharp object is calculated: P(injury) = 612 sharp object injuries / 56,971,309 hours worked
  • The result is: P(injury) = 1.07 x 10^-5
  • The relative frequency method is used to assign probabilities based on historical data.

Statistics: Confidence Limits and Probability

  • Confidence limits are used to estimate the average exposure of a population.
  • The formula for calculating the average concentration is: X = (∑x_i) / n
  • The formula for calculating the standard deviation is: s = sqrt((∑(x_i - X)^2) / (n - 1))
  • The 95% confidence limits are given by: X ± 1.96(s / sqrt(n))

CSP Question 04/12/2004: Exposure Assessment

  • An industrial hygienist is performing an exposure assessment for toluene in a shoe factory.
  • The TLV for toluene is 50 ppm.
  • The 95% confidence limits for the average employee exposure are: 8.9, 37.1

CSP Question 01/11/2007: Test Scores

  • The average test score for students taking a certain safety exam is 163.
  • The standard deviation of the test scores is 24.
  • The test scores are normally distributed.
  • The test score of the 5th percentile is: 116

CSP Question 03/03/2011: Spearman Rank Correlation

  • The Spearman Rank Correlation is used to calculate the correlation between two ranked variables.
  • The formula for calculating the Spearman Rank Correlation is: rs = 1 - (6∑D^2) / (n(n-1))
  • The Spearman Rank Correlation between Audit Rank and Performance Rank for the four sites is: -1.00

CSP Question 04/05/2012: Poisson Distribution

  • The Poisson distribution is used to estimate the probability of a certain number of events occurring in a fixed interval.
  • The formula for calculating the Poisson distribution is: P(x) = (λ^x * e^(-λ)) / x!
  • The probability of a school having four children diagnosed with a CNS cancer in one year is: 4.6 x 10^-10

CSP Question 11/15/2012: Injury Probability

  • The probability of an employee sustaining an injury due to contact with a sharp object within any one particular hour of operation is: 1.07 x 10^-5
  • The approach to assigning probabilities based on historical data is referred to as the "relative frequency method".

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Bowen EHS Inc. CSP Question 04/12/2004 - Statistics PDF

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