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Ch. 8 - Hypothesis Testing with Two Samples
Larson - Elementary Statistics: Picturing the World 8th Edition
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Not the one you use?Change textbook
All textbooksLarson 8th EditionCh. 8 - Hypothesis Testing with Two SamplesProblem 8.12b
Chapter 8, Problem 8.12b

The mean room rate for two adults for a random sample of 26 three-star hotels in Cincinnati has a sample standard deviation of \$31. Assume the population is normally distributed. (Adapted from Expedia)


Construct a 99% confidence interval for the population standard deviation.

Verified step by step guidance
1
Step 1: Recognize that the problem involves constructing a confidence interval for the population standard deviation. Since the population is normally distributed, we will use the chi-square distribution for this calculation.
Step 2: Identify the given values: the sample size (n = 26), the sample standard deviation (s = \$31), and the confidence level (99%).
Step 3: Determine the degrees of freedom (df) for the chi-square distribution. The formula is: df = n - 1. Substitute the sample size to calculate df.
Step 4: Find the critical chi-square values for the 99% confidence level. Use a chi-square table or statistical software to find the values for χ²_{α/2} (upper critical value) and χ²_{1-α/2} (lower critical value), where α = 1 - 0.99 = 0.01.
Step 5: Use the formula for the confidence interval of the population standard deviation: \(\sqrt{\frac{(n-1)s^2}{χ²_{α/2}\)}} \(\leq\) \(\sigma\) \(\leq\) \(\sqrt{\frac{(n-1)s^2}{χ²_{1-α/2}\)}}. Substitute the values for n, s, and the critical chi-square values to calculate the interval.

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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 a sample statistic, that is likely to contain the population parameter with a specified level of confidence. For example, a 99% confidence interval suggests that if we were to take many samples and construct intervals in the same way, approximately 99% of those intervals would contain the true population parameter.
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Introduction to Confidence Intervals

Sample Standard Deviation

The sample standard deviation is a measure of the amount of variation or dispersion in a set of sample data points. It quantifies how much the individual data points deviate from the sample mean. In this context, it is used to estimate the variability of room rates among the sampled hotels, which is crucial for constructing the confidence interval.
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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. When constructing confidence intervals for population variances or standard deviations, the Chi-Square distribution is applied, particularly when the population is normally distributed, as is the case in this question.
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Related Practice
Textbook Question

Find the critical value(s) for the alternative hypothesis, level of significance , and sample sizes and . Assume that the samples are random and independent, the populations are normally distributed, and the population variances are (a) equal

Ha:μ1≠μ2 , α=0.01 , n1=19 , n2=22

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Textbook Question

Testing the Difference Between Two Means (b) find the critical value(s) and identify the rejection region(s), Assume the samples are random and dependent, and the populations are normally distributed.

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Textbook Question

Testing the Difference Between Two Means, (b) find the critical value(s) and identify the rejection region(s), Assume the samples are random and dependent, and the populations are normally distributed.

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A researcher claims that injections of onabotulinumtoxinA reduce the number of days per month that chronic migraine sufferers have headaches. The table shows the number of days chronic migraine sufferers suffered migraines before and after using the treatment. At , α= 0.01 is there enough evidence to support the researcher’s claim? (Adapted from Journal of Headache and Pain)

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Textbook Question

Testing the Difference Between Two Means (c) calculate d̄ and Sd, Assume the samples are random and dependent, and the populations are normally distributed.

[APPLET] Passing Play Percentages The passing play percentages of 10 randomly selected NCAA Division 1A college football teams for home and away games in the 2020–2021 season are shown in the table. At , α=0.20 is there enough evidence to support the claim that passing play percentage is different for home and away games? (Source: TeamRankings)

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Textbook Question

Testing the Difference Between Two Means (b) find the critical value(s) and identify the rejection region(s), Assume the samples are random and dependent, and the populations are normally distributed.

[APPLET] Passing Play Percentages The passing play percentages of 10 randomly selected NCAA Division 1A college football teams for home and away games in the 2020–2021 season are shown in the table. At , α=0.20 is there enough evidence to support the claim that passing play percentage is different for home and away games? (Source: TeamRankings)

42
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Textbook Question

Find the critical value(s) for the alternative hypothesis, level of significance , and sample sizes and . Assume that the samples are random and independent, the populations are normally distributed, and the population variances are (a) equal .

Ha:μ1<μ2 , α=0.10 , n1=30 , n2=32

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