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Ch. 5 - Normal Probability Distributions
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. 5 - Normal Probability DistributionsProblem 5.5.30
Chapter 5, Problem 5.5.30

"Getting Physical The figure shows the results of a survey of U.S. adults ages 18 to 29 who were asked whether they participated in a sport. In the survey, 48% of the men and 23% of the women said they participate in sports. The most common sports are shown below. Use this information in Exercises 29 and 30.



You randomly select 300 U.S. women ages 18 to 29 and ask them whether they participate in at least one sport. Of the 72 who say yes, 50% say they participate in volleyball. How likely is this result? Do you think this sample is a good one? Explain your reasoning."

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Step 1: Identify the given data from the problem. The survey states that 23% of U.S. women ages 18 to 29 participate in sports. Out of the 300 women surveyed, 72 said they participate in sports, and 50% of these 72 women said they participate in volleyball.
Step 2: Calculate the expected number of women who participate in sports based on the population proportion. Multiply the total number of women surveyed (300) by the proportion of women who participate in sports (23%). This gives the expected number of women who participate in sports.
Step 3: Compare the observed number of women who participate in sports (72) to the expected number calculated in Step 2. Determine if the observed number significantly deviates from the expected number using statistical reasoning.
Step 4: Calculate the expected number of women who participate in volleyball based on the population proportion. Multiply the expected number of women who participate in sports (from Step 2) by the proportion of women who participate in volleyball (13%).
Step 5: Compare the observed number of women who participate in volleyball (50% of 72, which is 36) to the expected number calculated in Step 4. Use statistical reasoning to assess whether the sample is representative and whether the result is likely or unusual.

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Key Concepts

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

Sampling and Sample Size

Sampling refers to the process of selecting a subset of individuals from a population to estimate characteristics of the whole group. The sample size, in this case, 300 women, is crucial as larger samples tend to provide more reliable estimates of population parameters. A well-chosen sample can help ensure that the results are representative and minimize sampling bias.
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Probability and Likelihood

Probability is a measure of the likelihood that a particular event will occur, expressed as a number between 0 and 1. In this context, calculating the likelihood of 50% of the respondents participating in volleyball involves understanding how this figure compares to the overall participation rates for women in sports. This helps assess whether the observed result is statistically significant or could be due to random chance.
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Statistical Significance

Statistical significance indicates whether the results observed in a sample are likely to reflect true effects in the population or if they could have occurred by random chance. In evaluating the sample of 72 women, one must consider the expected participation rates and the sample size to determine if the finding of 50% participation in volleyball is a meaningful result or an anomaly.
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Related Practice
Textbook Question

Testing a Drug A drug manufacturer claims that a drug cures a rare skin disease 75% of the time. The claim is checked by testing the drug on 100 patients. If at least 70 patients are cured, then this claim will be accepted. Use this information in Exercises 31 and 32.


Find the probability that the claim will be accepted, assuming that the actual probability that the drug cures the skin disease is 65%.

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Textbook Question

In Exercises 21–24, a control chart is shown. Each chart has horizontal lines drawn at the mean mu, at mu ±2sigma, and at mu±3sigma. Determine whether the process shown is in control or out of control. Explain.


A gear has been designed to have a diameter of 3 inches. The standard deviation of the process is 0.2 inch.


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Textbook Question

Finding Probabilities for Sampling Distributions In Exercises 29–32, find the indicated probability and interpret the results.


Dow Jones Industrial Average From 1975 through 2020, the mean annual gain of the Dow Jones Industrial Average was 652. A random sample of 32 years is selected from this population. What is the probability that the mean gain for the sample was between 400 and 700? Assume sigma=1540

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Textbook Question

In Exercises 1–4, the sample size n, probability of success p, and probability of failure q are given for a binomial experiment. Determine whether you can use a normal distribution to approximate the distribution of x.

n=18, p=0.90, q=0.10

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Textbook Question

Paint Cans A machine is set to fill paint cans with a mean of 128 ounces and a standard deviation of 0.2 ounce. A random sample of 40 cans has a mean of 127.9 ounces. The machine needs to be reset when the mean of a random sample is unusual. Does the machine need to be reset? Explain.

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Textbook Question

Approximating a Binomial Distribution In Exercises 17 and 18, a binomial experiment is given. Determine whether you can use a normal distribution to approximate the binomial distribution. If you can, find the mean and standard deviation. If you cannot, explain why.


Bachelor’s Degrees Twenty-two percent of adults over 18 years of age have a bachelor’s degree. You randomly select 20 adults over 18 years of age and ask whether they have a bachelor’s degree.

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