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Ch. 7 - Estimating Parameters and Determining Sample Sizes
Triola - Elementary Statistics 14th Edition
Triola14th EditionElementary StatisticsISBN: 9780137366446Not the one you use?Change textbook
All textbooksTriola 14th EditionCh. 7 - Estimating Parameters and Determining Sample SizesProblem 7.3.16b
Chapter 7, Problem 7.3.16b

Comparing Waiting Lines


The values listed below are waiting times (in minutes) of customers at the Bank of Providence, where customers may enter any one of three different lines that have formed at three teller windows. Construct a 95% confidence interval for the population standard deviation sigma.
Customer waiting times at Bank of Providence: 4.2, 5.4, 5.8, 6.2, 6.7, 7.7, 7.7, 8.5, 9.3, 10.0 minutes.

Verified step by step guidance
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Step 1: Identify the sample data provided. The waiting times are: 4.2, 5.4, 5.8, 6.2, 6.7, 7.7, 7.7, 8.5, 9.3, and 10.0 minutes. Note that the sample size (n) is 10.
Step 2: Calculate the sample variance (s²). First, compute the sample mean (x̄) using the formula: x¯=∑xin. Then, use the formula for variance: s2=∑(xi-x¯)2n-1.
Step 3: Use the Chi-Square distribution to construct the confidence interval for the population standard deviation (σ). The formula for the confidence interval is: [(n-1)s2χ2,(n-1)s2χ1], where χ1 and χ2 are the critical values from the Chi-Square table for the given confidence level and degrees of freedom (df = n - 1).
Step 4: Look up the critical values for the Chi-Square distribution at a 95% confidence level with degrees of freedom (df = n - 1 = 9). Use a Chi-Square table or calculator to find these values.
Step 5: Compute the confidence interval for the population standard deviation (σ) by taking the square root of the lower and upper bounds of the variance confidence interval. The formula is: [(n-1)s2χ2,(n-1)s2χ1].

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Confidence Interval

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the population parameter with a specified level of confidence, typically expressed as a percentage. For example, a 95% confidence interval suggests that if we were to take many samples and construct intervals in the same way, approximately 95% of those intervals would contain the true population parameter.
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Introduction to Confidence Intervals

Standard Deviation

Standard deviation is a measure of the amount of variation or dispersion in a set of values. A low standard deviation indicates that the values tend to be close to the mean, while a high standard deviation indicates that the values are spread out over a wider range. In the context of waiting times, it helps to understand how consistent or variable the waiting times are for customers.
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Calculating Standard Deviation

Chi-Square Distribution

The chi-square distribution is a statistical distribution that is used to estimate the variance of a population based on sample data. It is particularly useful when constructing confidence intervals for the population standard deviation. The distribution is defined by degrees of freedom, which in this case is related to the sample size, and is critical for determining the critical values needed for the confidence interval calculation.
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Related Practice
Textbook Question

Online Gambling Some states now allow online gambling. As a marketing manager for a casino, you need to determine the percentage of adults in those states who gamble online. How many adults must you survey in order to be 99% confident that your estimate is in error by no more than two percentage points?


b. Assume that 18% of all adults gamble online (based on 2017 data from a Gambling Commission study in Great Britain).

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Textbook Question

Minting Quarters Listed below are weights (grams) of quarters minted after 1964 (based on Data Set 40 “Coin Weights” in Appendix B).


b. Specifications require that the quarters have a weight of 5.670 g. What does the confidence interval suggest about that specification?


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Textbook Question

7. FRESHMAN 15 Here is a sample of amounts of weight change (kg) of college students in their freshman year (from Data Set 13 “Freshman 15” in Appendix B): 11, 3, 0, –2, where –2 represents a loss of 2 kg and positive values represent weight gained. Here are ten bootstrap samples:

{11, 11, 11, 0}, {11, –2, 0, 11}, {11, –2, 3, 0}, {3, –2, 0, 11}, {0, 0, 0, 3}, {3, –2, 3, –2}, {11, 3, –2, 0}, {–2, 3, –2, 3}, {–2, 0, –2, 3}, {3, 11, 11, 11}.

b. Using only the ten given bootstrap samples, construct an 80% confidence interval estimate of the standard deviation of the weight changes for the population.

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Textbook Question

Critical Thinking. In Exercises 17–28, use the data and confidence level to construct a confidence interval estimate of p, then address the given question.


OxyContin The drug OxyContin (oxycodone) is used to treat pain, but it is dangerous because it is addictive and can be lethal. In clinical trials, 227 subjects were treated with OxyContin and 52 of them developed nausea (based on data from Purdue Pharma L.P.).


b. Compare the result from part (a) to this 95% confidence interval for 5 subjects who developed nausea among the 45 subjects given a placebo instead of OxyContin: . What do you conclude?

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Textbook Question

Alcohol in Children’s Movies Listed below is a simple random sample of times (seconds) that animated children’s movies showed the use of alcohol (based on Data Set 20 “Alcohol and Tobacco in Movies” in Appendix B).


b. Are the requirements for constructing a 95% confidence interval estimate of the population standard deviation satisfied? If so, construct that confidence interval.


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Textbook Question

Mean Pulse Rate of Males Data Set 1 “Body Data” in Appendix B includes pulse rates of 153 randomly selected adult males, and those pulse rates vary from a low of 40 bpm to a high of 104 bpm. Find the minimum sample size required to estimate the mean pulse rate of adult males. Assume that we want 99% confidence that the sample mean is within 2 bpm of the population mean.


b. Assume that sigma=11.3 bpm, based on the value of s=11.3 bpm for the sample of 153 male pulse rates.


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