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Chapter 10: Problem 10

Conduct the appropriate test. A simple random sample of size \(n=320\) adults was asked their favorite ice cream flavor. Of the 320 individuals surveyed, 58 responded that they preferred mint chocolate chip. Test the claim that less than \(25 \%\) of adults prefer mint chocolate chip ice cream at the \(\alpha=0.01\) level of significance.

Short Answer

Expert verified
Reject \( H_0 \). There is sufficient evidence to support the claim that less than \( 25 \% \) of adults prefer mint chocolate chip ice cream.

Step by step solution

01

- State the Hypotheses

Identify the null and alternative hypotheses.\( H_0: p \geq 0.25 \) \( H_a: p < 0.25 \)
02

- Identify Significance Level and Sample Proportion

The significance level is given as \( \alpha=0.01 \). Calculate the sample proportion: \( \hat{p} = \frac{58}{320} = 0.18125 \)
03

- Calculate the Test Statistic

Use the formula for the z-test for proportions:\[ z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0 (1 - p_0)}{n}}} \] Substituting the values \( p_0 = 0.25 \), \( \hat{p} = 0.18125 \), and \( n = 320 \), we get: \[ z = \frac{0.18125 - 0.25}{\sqrt{\frac{0.25 (1 - 0.25)}{320}}} = \frac{0.18125 - 0.25}{\sqrt{\frac{0.25 \cdot 0.75}{320}}} = \frac{-0.06875}{0.024215} \approx -2.84 \]
04

- Determine Critical Value and Make Decision

For a one-tailed test at \( \alpha = 0.01 \), the critical z-value is \( -2.33 \). Compare the test statistic to the critical value: \( z = -2.84 \) is less than \( -2.33 \). Since \( z \) is in the rejection region, reject \( H_0 \).
05

Conclusion

Since the null hypothesis is rejected, there is sufficient evidence at the \( \alpha = 0.01 \) level to support the claim that less than \( 25 \% \) of adults prefer mint chocolate chip ice cream.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Null Hypothesis
In hypothesis testing, the null hypothesis is the statement we aim to test. It represents a default position, suggesting that there is no effect or no difference. In the context of the given exercise, the null hypothesis (\( H_0 \)) is that at least 25% of adults prefer mint chocolate chip ice cream. We express this mathematically as:

\( H_0: p \geq 0.25 \)

This means we begin by assuming that the proportion of adults who prefer mint chocolate chip ice cream is 25% or higher.
Alternative Hypothesis
The alternative hypothesis (\( H_a \)) challenges the null hypothesis. It represents the outcome that the researcher expects or is trying to prove. In the given example, the alternative hypothesis is that less than 25% of adults prefer mint chocolate chip ice cream. This can be written as:

\( H_a: p < 0.25 \)

We use the alternative hypothesis to guide our decision-making process. If the data provide significant evidence, we reject the null hypothesis in favor of the alternative hypothesis.
Z-Test for Proportions
The z-test for proportions is a statistical method used to determine if a sample proportion significantly differs from a known population proportion. Here鈥檚 the formula we use:

\[ z = \frac{\hat{p} - p_0}{\frac{\sqrt{p_0 (1 - p_0)}}{n}} \]

where

\hat{p} is the sample proportion (in this case, 0.18125),
\( p_0 \) is the known population proportion (0.25),
and \( n \) is the sample size (320).

By substituting these values into the formula, we calculated the z-score to be approximately -2.84. This z-score tells us how many standard deviations the sample proportion is away from the population proportion.
Rejection Region
The rejection region is a critical part of hypothesis testing. It defines the range of values for which we reject the null hypothesis. For a one-tailed test at the 0.01 significance level, the rejection region lies in the lower tail of the normal distribution. The critical value for this level is -2.33.

Since the calculated z-score (-2.84) is less than the critical value (-2.33), our test statistic falls into the rejection region. Therefore, we reject the null hypothesis. This means there is significant evidence at the 0.01 level to support the claim that less than 25% of adults prefer mint chocolate chip ice cream.

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Most popular questions from this chapter

To test \(H_{0}: \mu=4.5\) versus \(H_{1}: \mu>4.5,\) a simple random sample of size \(n=13\) is obtained from a population that is known to be normally distributed. (a) If \(\bar{x}=4.9\) and \(s=1.3,\) compute the test statistic. (b) Draw a \(t\) -distribution with the area that represents the \(P\) -value shaded. (c) Approximate and interpret the \(P\) -value. (d) If the researcher decides to test this hypothesis at the \(\alpha=0.1\) level of significance, will the researcher reject the null hypothesis? Why?

According to the Statistical Abstract of the United States, in \(2000,10.7 \%\) of Americans over 65 years of age used the Internet. A researcher believes the proportion of Americans over 65 years of age who use the Internet is higher than \(10.7 \%\) today. (a) Determine the null and alternative hypotheses. (b) Suppose sample data indicate that the null hypothesis should be rejected. State the conclusion of the researcher. (c) Suppose, in fact, the percentage of Americans over 65 years of age who use the Internet is still \(10.7 \% .\) Was a Type I or Type II error committed?

The mean yield per acre of soybeans on farms in the United States in 2003 was 33.5 bushels, according to data obtained from the U.S. Department of Agriculture. A farmer in Iowa claimed the yield was higher than the reported mean. He randomly sampled 35 acres on his farm and determined the mean yield to be 37.1 bushels, with a standard deviation of 2.5 bushels. He computed the \(P\) -value to be less than 0.0001 and concluded that the U.S. Department of Agriculture was wrong. Why should his conclusions be looked on with skepticism?

State the requirements that must be satisfied to test hypotheses about a population mean with \(\sigma\) unknown.

In your own words, explain the difference between "beyond all reasonable doubt" and "beyond all doubt."
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