Business Statistics Chapter 8
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Questions and Answers

What is the population size in the example?

  • 100
  • 1,600 (correct)
  • 5,530
  • 1,480

What is the sample size in the given example?

  • 5,530
  • 1,600
  • 1,480
  • 100 (correct)

What is the value of the sample mean?

  • 5,530 (correct)
  • 1,600
  • 1,480
  • 100

What is the value of the sample standard deviation?

<p>1,480 (A)</p> Signup and view all the answers

What is the degrees of freedom (df) used to find the t-value?

<p>99 (B)</p> Signup and view all the answers

What is the significance level (α) used in calculating the 95% confidence interval?

<p>0.05 (A)</p> Signup and view all the answers

What is the t-value (tα/2) for a 95% confidence interval and df = 99?

<p>1.984 (A)</p> Signup and view all the answers

What is the approximate standard error of the mean calculated in the example?

<p>14.8 (D)</p> Signup and view all the answers

What does the significance level, α, indicate?

<p>The probability that a confidence interval will not contain the true population mean. (A)</p> Signup and view all the answers

What is the relationship between confidence level and significance level?

<p>Confidence level is the complement of significance level. (B)</p> Signup and view all the answers

Which of the following z-scores corresponds to a 90% confidence level?

<p>1.645 (A)</p> Signup and view all the answers

What happens to the confidence interval as the confidence level increases?

<p>The interval becomes wider. (A)</p> Signup and view all the answers

Which formula correctly represents the use of the CONFIDENCE.NORM function in Excel?

<p>=CONFIDENCE.NORM(alpha, standard_dev, size) (C)</p> Signup and view all the answers

If a confidence interval is 95% confident, what is the corresponding significance level?

<p>0.05 (D)</p> Signup and view all the answers

What is the significance level for a 99% confidence interval?

<p>0.01 (C)</p> Signup and view all the answers

What does an increase in the critical z-score imply about the confidence interval?

<p>It results in a wider confidence interval. (C)</p> Signup and view all the answers

What does the function T.INV.2T calculate in Excel?

<p>The critical t-score (B)</p> Signup and view all the answers

What parameter is NOT required for the CONFIDENCE.T function in Excel?

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

Under what conditions can the binomial distribution be approximated by the normal distribution?

<p>np ≥ 5 and n(1 - p) ≥ 5 (B)</p> Signup and view all the answers

In the formula for calculating the standard error of the proportion, what does 'p' represent?

<p>The population proportion (D)</p> Signup and view all the answers

How is the degrees of freedom calculated for using the T.INV.2T function?

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

What does the margin of error represent in the context of confidence intervals?

<p>The range within which the true population parameter lies (A)</p> Signup and view all the answers

Which of the following is NOT a component of the CONFIDENCE.T function in Excel?

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

If a sample size is 15 and the significance level is 0.05, what would be the degrees of freedom used in the T.INV.2T function?

<p>14 (B)</p> Signup and view all the answers

What does the sample proportion represent in the context of estimating the population proportion?

<p>An estimate of the population proportion based on sample data (D)</p> Signup and view all the answers

How does increasing the sample size affect the margin of error in a confidence interval?

<p>Decreases the margin of error (D)</p> Signup and view all the answers

What is the critical z-score for a 98% confidence interval?

<p>2.33 (D)</p> Signup and view all the answers

Which of the following is true about the margin of error in a confidence interval for proportions?

<p>It represents the range between the sample proportion and the population proportion. (C)</p> Signup and view all the answers

What formula is used to calculate the approximate standard error of the proportion?

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

What is the required sample size to estimate the population proportion of U.S. citizens with blue eyes within a margin of error of ± 5% at 95% confidence?

<p>264 (C)</p> Signup and view all the answers

In the confidence interval formula for a proportion, what does the term 'critical z-score' reflect?

<p>The z-value that corresponds to the desired confidence level (B)</p> Signup and view all the answers

What factor must be used when determining a confidence interval for a finite population, especially when sampling is without replacement?

<p>Finite population correction factor (A)</p> Signup and view all the answers

If a sample proportion is found to be 0.22, which of the following represents the upper limit of the 98% confidence interval?

<p>0.316 (D)</p> Signup and view all the answers

When should the finite population correction factor be applied in sample calculations?

<p>When n/N is greater than 0.05 (D)</p> Signup and view all the answers

What information is necessary to determine a confidence interval for proportions?

<p>Sample proportion, sample size, and critical z-score (C)</p> Signup and view all the answers

If a sample of 100 students has a mean of 5,530 steps per day and a standard deviation of 1,480 steps, what is the next step to calculate a 95% confidence interval?

<p>Calculate z-value and adjust for standard error (D)</p> Signup and view all the answers

What is the importance of rounding up when calculating the sample size needed?

<p>To avoid underestimating the required sample size (A)</p> Signup and view all the answers

What impact does sampling without replacement have when estimating the standard error?

<p>It overestimates the standard error (A)</p> Signup and view all the answers

What is the formula to calculate a confidence interval for the mean of a finite population when the population standard deviation is known?

<p>Mean ± (z * standard deviation) (C)</p> Signup and view all the answers

In the example of measuring steps per day, why is the ratio n/N significant?

<p>It indicates if the finite population correction is necessary (D)</p> Signup and view all the answers

What does a margin of error represent in a confidence interval?

<p>The width of the confidence interval around a sample mean. (C)</p> Signup and view all the answers

What happens to the margin of error if the sample size is increased while keeping the confidence level constant?

<p>The margin of error decreases, resulting in a narrower interval. (B)</p> Signup and view all the answers

Which statement best describes a confidence interval at a 90% confidence level?

<p>90% of all sample means will produce intervals that contain the population mean. (A)</p> Signup and view all the answers

If the sample mean is 145.5, what are the endpoints of the 90% confidence interval?

<p>137.10 and 153.90 (B)</p> Signup and view all the answers

What is a common misunderstanding regarding confidence intervals?

<p>There is a specific probability that a given interval contains the true mean. (D)</p> Signup and view all the answers

What does an increase in the confidence level lead to when constructing a confidence interval?

<p>A wider confidence interval (A)</p> Signup and view all the answers

How can one interpret the concept of '90% confidence' in a confidence interval?

<p>90% of constructed intervals from repeated samples will contain the population mean. (C)</p> Signup and view all the answers

Which of the following best defines a point estimate in this context?

<p>The mean of a sample used to approximate the population mean. (D)</p> Signup and view all the answers

Flashcards

Confidence Interval

A range of values that estimates a population parameter with a specified level of confidence.

Margin of Error (ME)

The range of values above and below the sample mean in a confidence interval.

Sample Mean

The average of a sample, used as an estimate of the population mean.

Increasing Sample Size

Larger sample sizes reduce the Margin of Error and create narrower confidence intervals.

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90% Confidence Level

We are 90% confident that the population mean lies within the confidence interval.

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Interpreting Confidence Intervals

Confident intervals indicate that a population mean is likely within the calculated range, based on samples.

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Population Mean

The average of the entire population, which we estimate using sample data.

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Interval Endpoints

The lower and upper limits of a confidence interval indicating the range of estimates.

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T.INV.2T function

Excel function to find critical t-scores for confidence intervals.

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Alpha (α)

The significance level used in confidence interval calculations.

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Degrees of freedom

Calculated as sample size minus one (n - 1).

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CONFIDENCE.T function

Excel function for calculating margin of error for mean confidence intervals.

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Population standard deviation

A measure of data spread in the entire population, unknown in some cases.

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Binomial distribution

A probability distribution for successes in a fixed number of trials.

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Sample proportion

Ratio of successes in a sample, denoted as p.

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Standard error of the proportion

The standard deviation of the sample proportion distribution.

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Significance Level (α)

The probability that a confidence interval does not contain the true population mean.

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Confidence Level

The complement of the significance level, represented as 1 - α.

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Confidence Interval Width

The range within which the true population mean is expected to fall, wider with higher confidence levels.

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Critical z-score (zα/2)

The z-value corresponding to a specific confidence level and significance level.

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95% Confidence Interval

A confidence interval that contains the true mean with 95% certainty, α = 0.05.

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Excel's CONFIDENCE.NORM

A function to calculate margin of error for confidence intervals when population standard deviation is known.

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Standard Deviation

A measure of the amount of variation or dispersion of a set of values.

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Sample Size

The number of observations or data points in a sample used for estimation.

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Sample Standard Deviation

A measure of the amount of variation in a sample's values.

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Standard Error of the Mean (SEM)

An estimate of the variability of the sample mean from the population mean.

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t-Value (tα/2)

A value from the t-distribution used for confidence interval calculations.

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Confidence Interval Calculation

A method to determine the range in which the population mean likely falls.

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Endpoint Calculation

Determining the lower and upper limits of the confidence interval.

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Population Size Impact

The total number of individuals in a defined group for study.

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Sample Size for Proportion

The number of observations needed to estimate a population proportion with a certain level of confidence and margin of error.

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Zα/2 for 95%

The critical value that corresponds to a 95% confidence level in statistical hypothesis testing.

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Finite Population Correction Factor

A factor used to adjust the standard error when sampling without replacement from a finite population.

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Confidence Interval for Mean (σ Known)

A range of values derived from a sample that is likely to contain the population mean, when population standard deviation is known.

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Confidence Interval for Mean (σ Unknown)

A range of values that estimates the population mean when the population standard deviation is unknown.

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Confidence Interval for Proportion

A range that estimates the population proportion based on sample data.

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Standard Error of Proportion

An estimate of the variability of the sample proportion, calculated using the formula SE = √(p̂(1-p̂)/n).

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Margin of Error

The range of uncertainty in sample estimates; the maximum expected difference between the population parameter and the sample estimate.

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Critical z-score

A value from the z-distribution that corresponds to the desired confidence level.

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Margin of Error

The amount that is allowed for in case of miscalculation or change in conditions, denoted as MEp.

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Calculating Confidence Intervals

The process includes finding sample proportion, standard error, critical z-score, and calculating endpoints.

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Example of Confidence Interval

Using the sample of 100 people, the 98% confidence interval for blue eyes proportion is [0.124, 0.316].

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Study Notes

Chapter 8

  • This chapter covers business statistics, specifically confidence intervals.
  • The book is titled "Business Statistics" and authored by Robert A. Donnelly, Jr., with contributions from Serina Al Haddad and Stefan Ruediger.
  • There are different sections within chapter 8, focusing on point estimates, calculating confidence intervals for the mean (with known and unknown standard deviation), calculating confidence intervals for proportions, determining sample size, and calculating confidence intervals for finite populations.
  • The Bureau of Labor Statistics (BLS) uses samples to estimate unemployment rates because the population is large.
  • A recent example shows that the unemployment rate may have increased from September 2017 to March 2018, but the actual rate was between 3.95% and 4.25% in March 2018 with a 90% confidence level.
  • There's a discussion of point estimates, their simplicity, and lack of information about accuracy, contrasted with interval estimates.

8.1 Point Estimates

  • A point estimate is a single value representing a population of interest, like a sample mean or proportion.
  • The sample mean estimates the unknown population mean, while the sample proportion estimates the population proportion.

8.2 Confidence Intervals for the Mean (σ Known)

  • A confidence interval is an interval estimate around a sample mean, defining a range where the true population mean is likely to lie.
  • A confidence level represents the probability that the interval estimate will include the population parameter.
  • Assumptions for this section are a sample size of at least 30, and the population standard deviation (σ) is known.
  • The formula for the standard error of the mean is given by σₓ= σ/ √n where σ = population standard deviation, and n = sample size.

8.3 Confidence Intervals for the Mean (σ Unknown)

  • When the population standard deviation isn't known, the sample standard deviation (s) is used instead.
  • The Student's t-distribution is used with the sample standard deviation and sample size.
  • The formula for standard error of the mean using sample sd (s) is ô = s/√n where s = sample standard deviation and n = sample size.
  • The shape of the t-distribution resembles the normal distribution as the degrees of freedom increase, which means a larger sample size is practically the same as the normal distribution.

8.4 Confidence Intervals for Proportions

  • Binomial distribution is used for proportion data.
  • These proportions are approximated using the normal distribution when np ≥ 5 and n(1-p) ≥ 5 where n= sample size and p= the probability of a success.
  • The formula for sample proportion is p = x/n where x = number of successes and n = sample size.
  • Standard error for the proportion is σ = √p(1-p)/n where p = sample proportion and n = sample size.
  • Formulas for confidence intervals for proportions are given.

8.5 Determining the Sample Size

  • Increasing the sample size reduces the margin of error leading to a more precise confidence interval.
  • To calculate a sample size with given confidence level and population standard deviation, certain formulas are used.

8.6 Calculating Confidence Interval for Finite Populations

  • When sampling from a finite population, the standard error calculation needs adjustment.
  • A factor to correct the standard error is included when n/N > 0.05 (n = sample size and N = population size) using the formula (N - n)/√N-1 where N = Population size and n = sample size

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Explore Chapter 8 of 'Business Statistics' by Robert A. Donnelly, Jr., which delves into confidence intervals. This chapter discusses point estimates, calculating confidence intervals for means and proportions, as well as determining sample sizes and handling finite populations. Gain insights into practical applications, such as estimating unemployment rates using statistical samples.

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