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Ch. 6 - Confidence Intervals
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. 6 - Confidence IntervalsProblem 6.2.41
Chapter 6, Problem 6.2.41

Tennis Ball Manufacturing A company manufactures tennis balls. When the balls are dropped onto a concrete surface from a height of 100 inches, the company wants the mean bounce height to be 55.5 inches. This average is maintained by periodically testing random samples of 25 tennis balls. If the t-value falls between and , then the company will be satisfied that it is manufacturing acceptable tennis balls. For a random sample, the mean bounce height of the sample is 56.0 inches and the standard deviation is 0.25 inch. Assume the bounce heights are approximately normally distributed. Is the company making acceptable tennis balls? Explain.

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Step 1: Identify the null hypothesis (H₀) and the alternative hypothesis (H₁). The null hypothesis is that the mean bounce height is 55.5 inches (H₀: μ = 55.5), and the alternative hypothesis is that the mean bounce height is not 55.5 inches (H₁: μ ≠ 55.5).
Step 2: Calculate the test statistic (t-value) using the formula: t = x̄ - μsn, where x̄ is the sample mean (56.0), μ is the population mean (55.5), s is the sample standard deviation (0.25), and n is the sample size (25).
Step 3: Determine the degrees of freedom (df) for the t-distribution. The degrees of freedom are calculated as df = n - 1, where n is the sample size (25).
Step 4: Compare the calculated t-value to the critical t-values for the given significance level (not provided in the problem, but typically 0.05 for a two-tailed test). If the calculated t-value falls within the range of the critical t-values, the null hypothesis is not rejected, and the company is manufacturing acceptable tennis balls. Otherwise, the null hypothesis is rejected.
Step 5: Conclude whether the company is manufacturing acceptable tennis balls based on the comparison in Step 4. If the t-value is within the acceptable range, the company is meeting its quality standards; otherwise, it is not.

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Key Concepts

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

Sampling Distribution

The sampling distribution is the probability distribution of a statistic obtained from a larger population, formed by taking multiple samples. In this case, the mean bounce height of the tennis balls is calculated from random samples of 25 balls. Understanding this concept is crucial because it allows us to determine how sample means vary and how they relate to the population mean.
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Sampling Distribution of Sample Proportion

Hypothesis Testing

Hypothesis testing is a statistical method used to make inferences about population parameters based on sample data. In this scenario, the company is likely testing a null hypothesis that the mean bounce height is equal to 55.5 inches against an alternative hypothesis. This process involves calculating a test statistic (like a t-value) and comparing it to critical values to decide whether to reject or fail to reject the null hypothesis.
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Step 1: Write Hypotheses

Confidence Intervals

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. In this context, the company can use the sample mean and standard deviation to construct a confidence interval for the mean bounce height. If this interval includes the target mean of 55.5 inches, it suggests that the tennis balls are acceptable.
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Introduction to Confidence Intervals
Related Practice
Textbook Question

In Exercise 35, would it be unusual for the population mean to be over \$1500? Explain.

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Textbook Question

Graphical Analysis In Exercises 9–12, use the values on the number line to find the sampling error.

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Textbook Question

Which statistic is the best unbiased estimator for μ?

a. s

b. xbar

c. the median

d. the mode

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Textbook Question

In Exercises 9–12, construct the indicated confidence intervals for (a) the population variance and (b) the population standard deviation . Assume the sample is from a normally distributed population.

c = 0.99, s^2 = 0.64, n = 7

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Textbook Question

In Exercises 5–8, find the critical value zc necessary to construct a confidence interval at the level of confidence c.

c = 0.97

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Textbook Question

In Exercises 13–16, find the margin of error for the values of c, σ and n.

c = 0.95, σ = 5.2, n = 30

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