Ch. 8 - Hypothesis Testing

Triola - Elementary Statistics 14th Edition

Triola14th EditionElementary StatisticsISBN: 9780137366446Not the one you use?Change textbook

Triola 14th Edition

Ch. 8 - Hypothesis Testing

Problem 8.2.3

Chapter 8, Problem 8.2.3

Statistical Literacy and Critical Thinking
In Exercises 1–4, use the results from a Hankook Tire Gauge Index survey of a simple random sample of 1020 adults. Among the 1020 respondents, 86% rated themselves as above average drivers. We want to test the claim that more than 3/4 of adults rate themselves as above average drivers.

Requirements Are the requirements of the hypothesis test all satisfied? Explain.

Verified step by step guidance

Step 1: Identify the population proportion (p) and the sample proportion (p̂). The claim is that more than 3/4 (or 0.75) of adults rate themselves as above-average drivers. From the survey, 86% (or 0.86) of the 1020 respondents rated themselves as above-average drivers. Thus, p = 0.75 and p̂ = 0.86.

Step 2: Verify the sample size requirement for a hypothesis test of proportions. The sample size must be large enough such that both np and n(1-p) are greater than or equal to 5. Calculate these values using n = 1020 and p = 0.75: np = 1020 × 0.75 and n(1-p) = 1020 × (1 - 0.75).

Step 3: Check if the sample is a simple random sample. The problem states that the survey was conducted using a simple random sample of 1020 adults, which satisfies this requirement.

Step 4: Confirm that the population is at least 10 times larger than the sample size. The population of adults is much larger than 10 × 1020, so this condition is satisfied.

Step 5: Conclude whether the requirements for the hypothesis test are satisfied. If all the above conditions are met (large sample size, simple random sample, and sufficiently large population), then the requirements for conducting the hypothesis test are satisfied.

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Key Concepts

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

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about a population based on sample data. It involves formulating a null hypothesis (H0) and an alternative hypothesis (H1), then using sample data to determine whether to reject H0. In this case, the null hypothesis would state that 75% or fewer adults rate themselves as above average drivers, while the alternative would claim that more than 75% do.

Simple Random Sample

A simple random sample is a subset of individuals chosen from a larger population, where each individual has an equal chance of being selected. This method helps ensure that the sample is representative of the population, reducing bias in the results. In the context of the survey, the sample of 1020 adults must be random to validly generalize the findings to the entire adult population.

Requirements for Hypothesis Testing

For hypothesis testing to be valid, certain requirements must be met, including the independence of observations, a sufficiently large sample size, and the normality of the sampling distribution. In this scenario, the sample size of 1020 is large enough to invoke the Central Limit Theorem, which suggests that the sampling distribution of the sample proportion will be approximately normal, provided the sample is random.

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06:21

Step 1: Write Hypotheses

Related Practice

Textbook Question

Test Statistics

In Exercises 9–12, refer to the exercise identified and find the value of the test statistic. (Refer to Table 8-2 to select the correct expression for evaluating the test statistic.)

Exercise 5 “Landline Phones”

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Textbook Question

Finding Critical Values of (chi)^2 For large numbers of degrees of freedom, we can approximate critical values of as follows:

(chi)^2 = (1/2)(z + sqrt(2k-1))

Here k is the number of degrees of freedom and z is the critical value(s) found from technology or Table A-2. In Exercise 12 “Spoken Words” we have df = 55, so Table A-4 does not list an exact critical value. If we want to approximate a critical value of (chi)^2 in the right-tailed hypothesis test with α = 0.01 and a sample size of 56, we let k =55 with z = 2.33 (or the more accurate value of z = 2.326348 found from technology). Use this approximation to estimate the critical value of for Exercise 12. How close is it to the critical value of (chi)^2 = 82.292 obtained by using Statdisk and Minitab?

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Textbook Question

Testing Claims About Variation

In Exercises 5–16, test the given claim. Identify the null hypothesis, alternative hypothesis, test statistic, P-value, or critical value(s), then state the conclusion about the null hypothesis, as well as the final conclusion that addresses the original claim. Assume that a simple random sample is selected from a normally distributed population.

Pulse Rates of Men A simple random sample of 153 men results in a standard deviation of 11.3 beats per minute (based on Data Set 1 “Body Data” in Appendix B). The normal range of pulse rates of adults is typically given as 60 to 100 beats per minute. If the range rule of thumb is applied to that normal range, the result is a standard deviation of 10 beats per minute. Use the sample results with a 0.05 significance level to test the claim that pulse rates of men have a standard deviation equal to 10 beats per minute; see the accompanying StatCrunch display for this test. What do the results indicate about the effectiveness of using the range rule of thumb with the “normal range” from 60 to 100 beats per minute for estimating in this case?

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Textbook Question

Testing Claims About Proportions

In Exercises 9–32, test the given claim. Identify the null hypothesis, alternative hypothesis, test statistic, P-value, or critical value(s), then state the conclusion about the null hypothesis, as well as the final conclusion that addresses the original claim. Use the P-value method unless your instructor specifies otherwise. Use the normal distribution as an approximation to the binomial distribution, as described in Part 1 of this section.

Medical Malpractice In a study of 1228 randomly selected medical malpractice lawsuits, it was found that 856 of them were dropped or dismissed (based on data from the Physicians Insurers Association of America). Use a 0.01 significance level to test the claim that most medical malpractice lawsuits are dropped or dismissed. Should this be comforting to physicians?

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Textbook Question

Testing Claims About Proportions

Online Friends A Pew Research Center poll of 1060 teens aged 13 to 17 showed that 57% of them have made new friends online. Use a 0.01 significance level to test the claim that half of all teens have made new friends online.

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Textbook Question

Identifying H0 and H1

In Exercises 5–8, do the following:

a. Express the original claim in symbolic form.

b. Identify the null and alternative hypotheses.

Landline Phones Claim: Fewer than 10% of homes have only a landline telephone and no wireless phone. Sample data: A survey by the National Center for Health Statistics showed that among 16,113 homes, 5.8% had landline phones without wireless phones.

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