Statistics Chapter 5: Measures of Central Tendency
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Questions and Answers

What is the range of the repair cost?

  • $1
  • $2 (correct)
  • $0.50
  • $3

What is the correct sample variance for the weights provided?

  • 0.00156 grams²
  • 0.00389 grams² (correct)
  • 0.00225 grams²
  • 0.00400 grams²

What is the sample standard deviation for the weights of the candies?

  • 0.062 grams
  • 0.061 grams
  • 0.071 grams
  • 0.067 grams (correct)

Which histogram correctly represents the distribution of weights?

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

How is the distribution of the weights characterized?

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

What is the mean weight of the candies?

<p>0.843 grams (D)</p> Signup and view all the answers

Which measure of central tendency is most appropriate for the weight of the candy?

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

What can be concluded if the plotted values are not linear?

<p>The relationship represented by the plotted values cannot be analyzed using linear regression. (C)</p> Signup and view all the answers

What does the boxplot indicate about outliers if there are no points outside of the 1.5(IQR) boundary?

<p>There are no outliers present in the data. (B)</p> Signup and view all the answers

When constructing a 95% confidence interval to estimate the population mean cost, what is the implication of rounding to one decimal place?

<p>The confidence interval will be less precise but simpler to interpret. (D)</p> Signup and view all the answers

If a simple random sample of size n = 10 is taken and variability in repair costs is evenly distributed, what would happen to the width of a 95% confidence interval compared to a larger sample?

<p>It would be wider due to a smaller sample size. (C)</p> Signup and view all the answers

What can be said about the interpretation of a 95% confidence interval that is particularly wide?

<p>It suggests high variability in the sample data used for estimation. (A)</p> Signup and view all the answers

What happens to the confidence level if we were to calculate a 99% confidence interval instead of a 95% confidence interval?

<p>It would widen as we desire more certainty. (B)</p> Signup and view all the answers

When determining if there are outliers, what criterion is used if there are values beyond the first or third quartiles?

<p>Values beyond $1.5 * IQR$. (B)</p> Signup and view all the answers

Why might a second random sample from the same population yield different results for the estimated mean cost of repair?

<p>Sampling error could introduce variance. (D)</p> Signup and view all the answers

What is the point estimate for the population proportion of adults who believe that televisions are a luxury?

<p>0.498 (B)</p> Signup and view all the answers

Which requirement must be satisfied to construct a confidence interval for the population proportion?

<p>10 (D)</p> Signup and view all the answers

In a 95% confidence interval for the proportion who believe televisions are a luxury, which statement is correct?

<p>We are 95% confident the proportion lies within the interval. (B)</p> Signup and view all the answers

If the 95% confidence interval for the luxury belief is found to be (0.450, 0.546), what does this suggest about a supermajority believing so?

<p>It is highly unlikely since the upper bound is less than 0.60. (A)</p> Signup and view all the answers

What is the interpretation of a confidence interval result of (0.450, 0.546)?

<p>We expect the true proportion of luxury-believers to fall within this range 95% of the time. (A)</p> Signup and view all the answers

What is the implication of the population proportion being estimated at 0.498?

<p>Around half of the adults consider televisions a luxury. (D)</p> Signup and view all the answers

How does the sample size of 1029 affect the confidence interval's width?

<p>A larger sample size decreases the width of the confidence interval. (C)</p> Signup and view all the answers

What is needed to determine the standard error for the proportion in constructing a confidence interval?

<p>Sample proportion and sample size. (B)</p> Signup and view all the answers

What is the total number of entries in the data table?

<p>11 (C)</p> Signup and view all the answers

Which weight corresponds to the maximum total length recorded in the data table?

<p>155 kg (A)</p> Signup and view all the answers

For a sample size of 15, what is the critical value for the correlation coefficient?

<p>0.514 (B)</p> Signup and view all the answers

If the correlation coefficient is less than -0.514 for a sample size of 15, what can be concluded?

<p>The correlation is significant and negative. (A)</p> Signup and view all the answers

What does a larger z-score indicate for the weight of a baby born in week 34 compared to a baby born in week 40?

<p>The baby born in week 34 weighs relatively more. (A)</p> Signup and view all the answers

Which of the following total lengths appears most frequently in the data table?

<p>150.0 cm (B)</p> Signup and view all the answers

What is the correct interpretation of the first quartile (Q1)?

<p>25% of the monthly returns fall below or equal to Q1. (A)</p> Signup and view all the answers

What range of weights is represented in the data table?

<p>60 kg to 155 kg (B)</p> Signup and view all the answers

According to the correlation coefficients, what value would represent a weak correlation for a sample size of 12?

<p>0.388 (C)</p> Signup and view all the answers

Which of the following describes the second quartile (Q2)?

<p>Q2 is equivalent to the median of the dataset. (D)</p> Signup and view all the answers

What does the third quartile (Q3) represent in terms of the dataset?

<p>75% of the monthly returns fall below or equal to Q3. (A)</p> Signup and view all the answers

How many entries in the data table have a weight of 105 kg?

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

What is the minimum total length recorded in the data table?

<p>120.5 cm (B)</p> Signup and view all the answers

Which statement correctly describes outliers according to the given data?

<p>Outliers represent values that are significantly different from the rest of the data. (A)</p> Signup and view all the answers

What critical value would indicate that a correlation is significant for a sample size of 25?

<p>0.396 (C)</p> Signup and view all the answers

Which of the following statements is not true regarding the interpretation of quartiles?

<p>The third quartile indicates that 25% of the data is above it. (A)</p> Signup and view all the answers

Which option correctly represents the role of quartiles in identifying outliers?

<p>Outliers can only exist outside the interquartile range. (C)</p> Signup and view all the answers

What is the main purpose of determining quartiles in data analysis?

<p>To divide data into equal parts for analyzing distribution. (A)</p> Signup and view all the answers

How many pregnancies would be expected to last less than 210 days if 100 pregnant individuals were selected from the population?

<p>36 (A)</p> Signup and view all the answers

What is the expected number of samples with a mean of 210 days or less when selecting samples of size n = 18 from the population?

<p>6 (C)</p> Signup and view all the answers

If 100 independent samples of size n = 33 were obtained, how many samples would you expect to have a sample mean of 210 days or less?

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

What is the expected number of different students who would read more than 94 words per minute from a sample of 100?

<p>27 (A)</p> Signup and view all the answers

What is the conclusion about a mean reading rate of 90.1 wpm given that its probability is 0.1679?

<p>It is a typical result from the population. (D)</p> Signup and view all the answers

What is the confidence interval for the population mean number of books read?

<p>12.34 to 14.06 (C)</p> Signup and view all the answers

In the context of repair costs, what conclusion can be drawn regarding the plotted values?

<p>The plotted values show a clear linear pattern. (D)</p> Signup and view all the answers

What is the upper bound of the confidence interval for the mean cost of repair?

<p>$2484.3 (C)</p> Signup and view all the answers

Flashcards

Correlation Coefficient Critical Values

Values used to determine if observed correlation is statistically significant given sample size.

Correlation Coefficient Significance

A measure of the strength and direction of a linear relationship between two variables.

Sample Size (n)

The number of data points in a sample.

Critical Value (Correlation)

Specific value at which a relationship between variables is deemed statistically significant.

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Correlation Strength

The magnitude of association between variables.

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Data Table

An organized display of data in rows and columns, useful for comparison and analysis.

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Correlation Coefficient

A statistical measure that shows the degree and direction of correlation between two data sets.

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Positive Correlation

A relationship where both variables tend to increase or decrease together.

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Negative Correlation

A relationship where one variable tends to increase as the other decreases.

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Statistical Significance

A measure of how likely it is that an observed relationship between variables is not due to chance.

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Range of repair costs

The difference between the highest and lowest repair cost.

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Sample Variance (s^2)

A measure of how spread out numbers are from the average (mean).

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Sample Standard Deviation (s)

The square root of the sample variance. It's another way to measure how spread out the numbers are.

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Candy Weight Distribution

Describes the pattern of how candy weights are spread out.

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Frequency Histogram

A graph that shows the frequency or count of data points in specific ranges.

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Skewed Right Histogram

A histogram where the tail extends longer to the right

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Mean

The average of a set of numbers.

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Median

The middle value when data is ordered from least to greatest.

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Linear Relationship?

Whether the plotted data points show a straight line trend.

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Boxplot Outliers

Data points that are significantly far from the rest of the data, identified by being outside the 1.5 * IQR boundary.

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Confidence Interval

A range of values that is likely to contain the true population parameter (e.g., mean) with a certain level of confidence.

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95% Confidence Level

Indicates that we are 95% confident that the true population mean falls within the calculated confidence interval.

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Confidence Interval: Bigger Sample Size

A larger sample size generally leads to a narrower confidence interval, meaning more precise estimation of the population parameter.

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Variability in Data

The extent to which individual data points differ from each other and the overall trend.

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Confidence Interval: More Variability

Higher variability in the data tends to result in a wider confidence interval, as it becomes less certain to capture the true population parameter.

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Narrower Confidence Interval

A narrower confidence interval indicates a higher level of precision in estimating the population parameter.

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Z-score of a baby's weight at week 34

A standardized value that tells how many standard deviations an observation (in this case, the baby's weight at week 34) is from the mean. A larger z-score means the weight is further from the average, in a positive direction. A smaller z-score indicates the weight is closer to the mean.

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Interpretation of Quartiles (Q1, Q2, Q3)

Quartiles divide the sorted data into four equal parts. Q1 is the 25th percentile; Q2 the 50th, and Q3 the 75th. They show the spread of values, with Q1 defining the lower boundary of the middle 50% of values, Q3 outlining the upper boundary of the middle 50%, and Q2 representing the median.

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First Quartile (Q1)

The value below which 25% of the data falls.

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Second Quartile (Q2)

The median, the middle value when the data is ordered.

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Third Quartile (Q3)

The value below which 75% of the data falls.

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Outlier

A data point that significantly deviates from other observations in a data set.

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Monthly rate of return

Measures the change in the price of a stock or other investment each month, expressed as a percentage.

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Interpretation of quartiles: option C

Option C - 25% of the monthly returns are less than or equal to the first quartile; 50% are less than or equal to the second quartile; and 75% are less than or equal to the third quartile.

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Point Estimate

A single value used to estimate an unknown population parameter, based on a sample.

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Sample Proportion (p)

The proportion of individuals in a sample who possess a specific characteristic.

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Confidence Interval (CI)

A range of values that is likely to contain the true population parameter with a certain level of confidence.

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Confidence Level

The probability that a confidence interval will contain the true population parameter.

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Supermajority

A majority that exceeds a simple majority, often 60% or more.

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Requirement for CI: Sample Size

For a valid confidence interval, the sample size must be large enough.

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Requirement for CI: Random Sample

The sample must be randomly selected from the population to reflect the true population characteristics.

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Interpreting Confidence Interval

We are confident that the true population parameter lies within the calculated interval, based on the chosen confidence level.

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What is the expected number of pregnancies lasting less than 210 days if 100 are selected?

Given a population of pregnant individuals, we anticipate roughly 36 pregnancies out of 100 to have a duration shorter than 210 days.

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Sample Mean vs. Population Mean

The sample mean represents the average of a smaller group taken from a larger population. The population mean is the average of the entire population.

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How does sample size affect probability?

Increasing the sample size generally decreases the probability of extreme values. This is because larger samples better represent the population.

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What is a confidence interval?

A range of values where we are confident the true population mean lies, based on a sample.

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What is the confidence interval for the mean number of books read?

We are 90% confident that the true population mean number of books read is between 12.34 and 14.06.

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Linear Regression

A statistical technique to find a straight line that best represents the relationship between two variables.

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Outlier Test

A method to identify data points that are unusually far away from other values.

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Confidence Interval for Repair Cost

The range of values where we are 95% confident the true mean repair cost lies.

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Study Notes

  • Observational studies measure the response variable without influencing the explanatory variable.
  • Designed experiments assign individuals to groups, change the explanatory variable, and record the response variable.
  • Designed experiments allow researchers to claim causation between variables.
  • Observational studies do not allow researchers to claim causation.
  • In an experiment, the researcher controls one variable to determine its effect on another variable.
  • Observational studies only observe/record the value of response variable.
  • A statistical summary of a sample is called a statistic.
  • A statistical summary of a population is called a parameter.
  • A quantitative variable is discrete if it can be counted (e.g., number of books).
  • A quantitative variable is continuous if it cannot be counted (e.g., weight).
  • The range of a data set is the difference between the largest and smallest values.
  • Sample variance is a measure of the spread of data around the mean.
  • Sample standard deviation is the square root of sample variance.
  • The linear correlation coefficient (r) measures the strength and direction of a linear association between two variables.
  • A positive correlation means that higher values of one variable tend to be associated with higher values of the other variable.
  • A negative correlation means that higher values of one variable tend to be associated with lower values of the other variable.
  • A scatter diagram displays the relationship between two variables.
  • A linear correlation coefficient of 1 or -1 indicates a perfect linear relationship.
  • Values of r close to 0 indicate a weak linear relationship.
  • Critical values of r determine if a linear relationship exists between two variables.
  • A p-value is the probability of observing a result as extreme or more extreme than the observed result, assuming the null hypothesis is true.
  • The null hypothesis is the statement that there is no effect or relationship between variables being studied.
  • The alternate hypothesis is the statement that there is an effect or relationship between variables being studied.

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Description

This quiz covers key concepts from Chapter 5 of statistics, focusing on measures of central tendency, variance, standard deviation, and confidence intervals. Evaluate your understanding of distributions, outliers, and implications of sample sizes. Test your knowledge on how to interpret various statistical representations related to repair costs and weights of candies.

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