Ch. 8 - Hypothesis Testing

Triola - Elementary Statistics 14th Edition

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Triola 14th Edition

Ch. 8 - Hypothesis Testing

Problem 8.RE.4

Chapter 8, Problem 8.RE.4

Perception and Reality In a presidential election, 308 out of 611 voters surveyed said that they voted for the candidate who won (based on data from ICR Survey Research Group). Use a 0.05 significance level to test the claim that among all voters, the percentage who believe that they voted for the winning candidate is equal to 43%, which is the actual percentage of votes for the winning candidate. What does the result suggest about voter perceptions?

Verified step by step guidance

Step 1: Identify the null and alternative hypotheses. The null hypothesis (H₀) states that the percentage of voters who believe they voted for the winning candidate is equal to 43% (p = 0.43). The alternative hypothesis (H₁) states that the percentage is not equal to 43% (p ≠ 0.43).

Step 2: Calculate the test statistic using the formula for a proportion z-test:

z = (p - p

₀)p₀(1-p₀)n, where p is the sample proportion, p₀ is the hypothesized proportion, and n is the sample size.

Step 3: Compute the sample proportion (p) using the formula

\frac{x}{n}

, where x is the number of voters who said they voted for the winning candidate (308) and n is the total number of voters surveyed (611).

Step 4: Determine the critical z-value for a two-tailed test at a significance level of 0.05. Look up the z-value corresponding to 0.025 in each tail (since 0.05 is split between two tails) in a standard normal distribution table.

Step 5: Compare the calculated z-test statistic to the critical z-value. If the test statistic falls outside the range defined by the critical z-values, reject the null hypothesis. Otherwise, fail to reject the null hypothesis. Interpret the result in the context of voter perceptions.

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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 inferences 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 the percentage of voters who believe they voted for the winning candidate is 43%, while the alternative would suggest it is not.

Significance Level

The significance level, often denoted as alpha (α), is the threshold for determining whether a result is statistically significant. A common significance level is 0.05, which indicates a 5% risk of concluding that a difference exists when there is none. In this scenario, if the p-value from the hypothesis test is less than 0.05, it would suggest that the observed voter perception significantly differs from the claimed percentage.

P-Value

The p-value is a statistical measure that helps determine the strength of the evidence against the null hypothesis. It represents the probability of observing the sample data, or something more extreme, if the null hypothesis is true. A low p-value (typically less than the significance level) indicates strong evidence against the null hypothesis, suggesting that the actual voter perception may differ from the claimed 43%.

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Step 3: Get P-Value

Related Practice

Textbook Question

Hypothesis Test for Lightning Deaths Refer to the sample data given in Cumulative Review Exercise 1 and consider those data to be a random sample of annual lightning deaths from recent years. Use those data with a 0.01 significance level to test the claim that the mean number of annual lightning deaths is less than the mean of 72.6 deaths from the 1980s. If the mean is now lower than in the past, identify one of the several factors that could explain the decline.

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Textbook Question

Finding Critical Values

In Exercises 17–20, refer to the information in the given exercise and use a 0.05 significance level for the following.

a. Find the critical value(s).

b. Should we reject H0 or should we fail to reject H0?

Exercise 16

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Textbook Question

Job Search A Gallup poll of 195,600 employees showed that 51% of them were actively searching for new jobs. Use a 0.01 significance level to test the claim that the majority of employees are searching for new jobs

114

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Textbook Question

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.

Number and Proportions

a. Identify the actual number of respondents who rated themselves as above average drivers.

106

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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.

Light Year Claim: Most adults know that a light year is a measure of distance. Sample data: A Pew Research Center survey of 3278 adults showed that 72% knew that a light year is a measure of distance.

142

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Textbook Question

Type I Error and Type II Error

a. In general, what is a type I error? In general, what is a type II error?

364

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