Textbook Question
In Exercises 3–6, construct the indicated confidence interval for the population mean . Which distribution did you use to create the confidence interval?
c=0.90, x̅=8.21, σ=0.62, n=8
Ch. 8 - Hypothesis Testing with Two Samples
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470
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Table of contents
- Ch. 1 - Introduction to Statistics87
- Ch. 2 - Descriptive Statistics176
- Ch. 3 - Probability184
- Ch. 4 - Discrete Probability Distributions102
- Ch. 5 - Normal Probability Distributions183
- Ch. 6 - Confidence Intervals154
- Ch. 7 - Hypothesis Testing with One Sample165
- Ch. 8 - Hypothesis Testing with Two Samples134
- Ch. 9 - Correlation and Regression87
- Ch. 10 - Chi-Square Tests and the F-Distribution87
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Not the one you use?Change textbook
Chapter 8, Problem 8.CR.12a
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 variance.
Verified step by step guidance1
Step 1: Understand the problem. We are tasked with constructing a 99% confidence interval for the population variance based on a sample of 26 hotels. The sample standard deviation is given as \$31, and the population is assumed to be normally distributed. This means we can use the chi-square distribution to calculate the confidence interval.
Step 2: Recall the formula for the confidence interval for the population variance. The formula is: \[ \left( \frac{(n-1)s^2}{\chi^2_{\text{upper}}}, \frac{(n-1)s^2}{\chi^2_{\text{lower}}} \right) \] where \( n \) is the sample size, \( s^2 \) is the sample variance, and \( \chi^2_{\text{upper}} \) and \( \chi^2_{\text{lower}} \) are the critical values of the chi-square distribution corresponding to the desired confidence level.
Step 3: Calculate the sample variance \( s^2 \). The sample variance is the square of the sample standard deviation: \( s^2 = 31^2 \). This will be used in the formula.
Step 4: Determine the degrees of freedom \( df \). The degrees of freedom for the chi-square distribution is \( n-1 \), where \( n \) is the sample size. Here, \( df = 26 - 1 = 25 \).
Step 5: Find the critical chi-square values for a 99% confidence interval. Using a chi-square table or statistical software, locate \( \chi^2_{\text{upper}} \) and \( \chi^2_{\text{lower}} \) for \( df = 25 \) and a 99% confidence level. Plug these values, along with \( (n-1)s^2 \), into the formula to compute the confidence interval for the population variance.
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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, that is likely to contain the population parameter with a specified level of confidence. In this case, a 99% confidence interval means 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 variance.
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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. It quantifies how much the individual data points deviate from the sample mean. In this scenario, the sample standard deviation of $31 will be used to estimate the population variance, which is the square of the standard deviation.
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08:45Calculating Standard Deviation
Chi-Squared Distribution
The chi-squared distribution is a statistical distribution that is used to estimate the variance of a population based on sample data. When constructing confidence intervals for variance, the chi-squared distribution is applied, particularly when the population is normally distributed, as is the case here with the three-star hotels.
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07:01Intro to Least Squares Regression
Related Practice
Textbook Question
[APPLET] The annual earnings (in dollars) for 30 randomly selected locksmiths are shown below. Assume the population is normally distributed. (Adapted from Salary.com)
48,69446,85642,91261,67271,11254,861
69,45471,84159,75169,61254,28452,166
66,36048,16465,27235,25061,12765,397
58,92558,91659,01753,07045,19969,941
69,49257,08553,82952,69268,29853,792
Construct a 95% confidence interval for the population mean annual earnings for locksmiths.
Textbook Question
In Exercises 7–10, the statement represents a claim. Write its complement and state which is Ho and which is Ha.
σ=0.63
Textbook Question
Take this quiz as you would take a quiz in class. After you are done, check your work against the answers given in the back of the book.For each exercise, perform the steps below.
a. Identify the claim and state Ho and Ha
b. Determine whether the hypothesis test is left-tailed, right-tailed, or two-tailed, and whether to use a z-test or a t-test. Explain your reasoning.
c. Find the critical value(s) and identify the rejection region(s).
d. Find the appropriate standardized test statistic.
e. Decide whether to reject or fail to reject the null hypothesis.
f. Interpret the decision in the context of the original claim.
[APPLET] The table shows the credit scores for 12 randomly selected adults who are considered high-risk borrowers before and two years after they attend a personal finance seminar. At α=0.01, is there enough evidence to support the claim that the personal finance seminar helps adults increase their credit scores? Assume the populations are normally distributed.
Textbook Question
Take this quiz as you would take a quiz in class. After you are done, check your work against the answers given in the back of the book.For each exercise, perform the steps below.
a. Identify the claim and state Ho and Ha
b. Determine whether the hypothesis test is left-tailed, right-tailed, or two-tailed, and whether to use a z-test or a t-test. Explain your reasoning.
c. Find the critical value(s) and identify the rejection region(s).
d. Find the appropriate standardized test statistic.
e. Decide whether to reject or fail to reject the null hypothesis.
f. Interpret the decision in the context of the original claim.
A music teacher claims that the mean scores on a music assessment test for eighth grade students in public and private schools are equal. The mean score for 13 randomly selected public school students is 146 with a standard deviation of 49, and the mean score for 15 randomly selected private school students is 160 with a standard deviation of 42. At α=0.1, can you reject the teacher’s claim? Assume the populations are normally distributed and the population variances are equal. (Adapted from National Center for Education Statistics)
Textbook Question
[APPLET] The annual earnings (in dollars) for 30 randomly selected locksmiths are shown below. Assume the population is normally distributed. (Adapted from Salary.com)
48,69446,85642,91261,67271,11254,861
69,45471,84159,75169,61254,28452,166
66,36048,16465,27235,25061,12765,397
58,92558,91659,01753,07045,19969,941
69,49257,08553,82952,69268,29853,792
A researcher claims that the mean annual earnings for locksmiths is \$55,000. At α=0.05, can you reject the researcher’s claim? Interpret the decision in the context of the original claim.