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Chapter 9: Problem 27

Based on information from Harper's Index, \(r_{1}=\) 37 people out of a random sample of \(n_{1}=100\) adult Americans who did not attend college believe in extraterrestrials. However, out of a random sample of \(n_{2}=100\) adult Americans who did attend college, \(r_{2}=47\) claim that they believe in extraterrestrials. Does this indicate that the proportion of people who attended college and who believe in extraterrestrials is higher than the proportion who did not attend college? Use \(\alpha=0.01\).

Short Answer

Expert verified
There is not enough evidence to indicate a higher proportion for college attendees.

Step by step solution

01

Identify Variables and Hypotheses

We are comparing two proportions: those who did not attend college and those who did. Let \( p_1 \) be the proportion of non-college attendees who believe in extraterrestrials, and \( p_2 \) be the proportion of college attendees who believe. The null hypothesis \( H_0 \) is that \( p_1 = p_2 \), and the alternative hypothesis \( H_a \) is that \( p_1 < p_2 \).
02

Calculate Sample Proportions

Calculate the sample proportions for both groups: \( \hat{p}_1 = \frac{r_1}{n_1} = \frac{37}{100} = 0.37 \) and \( \hat{p}_2 = \frac{r_2}{n_2} = \frac{47}{100} = 0.47 \).
03

Calculate Pooled Proportion

The pooled proportion \( \hat{p} \) is used to estimate the common proportion given the null hypothesis: \[ \hat{p} = \frac{r_1 + r_2}{n_1 + n_2} = \frac{37 + 47}{100 + 100} = 0.42 \]
04

Calculate Standard Error

The standard error for the difference in proportions is calculated as: \[ SE = \sqrt{ \hat{p} (1 - \hat{p}) \left( \frac{1}{n_1} + \frac{1}{n_2} \right) } = \sqrt{ 0.42 \times 0.58 \times \left( \frac{1}{100} + \frac{1}{100} \right) } \approx 0.069 \]
05

Calculate Z-score

The Z-score is calculated using the formula: \[ Z = \frac{\hat{p}_2 - \hat{p}_1}{SE} = \frac{0.47 - 0.37}{0.069} \approx 1.45 \]
06

Determine Critical Value and Decision

At \( \alpha = 0.01 \), the critical value for a one-tailed test is approximately 2.33 from the Z-table. Since the calculated Z-score (1.45) is less than 2.33, we do not reject the null hypothesis.

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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, often represented as \( H_0 \), is a statement of no effect or no difference. It serves as a baseline or default position that we aim to challenge through our test. In our exercise, the null hypothesis states that the proportion of non-college attendees who believe in extraterrestrials is the same as that of college attendees. Mathematically, this can be expressed as \( p_1 = p_2 \).
  • If the test results support the null hypothesis, it suggests that any observed difference in the data could be due to random chance.
  • Failing to reject the null hypothesis does not prove that it is true; it merely indicates insufficient evidence to support the alternative hypothesis.
The null hypothesis is crucial because it provides a statement to test against. By determining whether evidence allows us to reject this assumption, we gain insights into whether any true difference exists between the groups studied.
Alternative Hypothesis
Standing in opposition to the null is the alternative hypothesis, denoted as \( H_a \). This hypothesis proposes that there is indeed a difference or effect, specifically tailored to the context of the research question. In our scenario, the alternative hypothesis asserts that the proportion of college attendees who believe in extraterrestrials (2) is greater than those who never attended college.
  • It is expressed mathematically as \( p_1 < p_2 \), suggesting a higher belief rate among college attendees.
  • The alternative hypothesis forms the basis of a directional test. Therefore, we use a one-tailed test approach to check if this predicted direction holds true.
Validated acceptance of the alternative hypothesis implies that the differences observed are statistically significant and unlikely the result of random sampling variability.
Proportions
Proportions are a type of statistic that represents a part of a total relationship, often expressed as a percentage or a fraction. In hypothesis testing involving categorical data, proportions help to convert raw counts into interpretable measures. For example:
  • The non-college group has a proportion of \( 7_1 = \frac{37}{100} = 0.37 \).
  • Meanwhile, the college group yields a proportion of \( 7_2 = \frac{47}{100} = 0.47 \).
By comparing these proportions, we aim to deduce if there is a tangible difference between the groups. Furthermore, the pooled proportion \( \hat{p} = 0.42 \) serves as an estimate for a common proportion under the null hypothesis, facilitating further calculations such as the standard error. Understanding these foundational elements allows for accurate statistical assessments of group differences.
Z-score
A Z-score is a statistical measure that indicates the number of standard deviations a data point is from the mean of the data set. In hypothesis testing, the Z-score is utilized to determine the position of a sample mean (or proportion) concerning the null hypothesis. Here鈥檚 how it鈥檚 applied in our problem:
  • Derived as \( Z = \frac{p_2 - p_1}{SE} \), where \( SE \) is the standard error of the difference in proportions.
  • Our calculated Z-score of approximately 1.45 expresses how many standard errors the observed difference of 0.10 (0.47 - 0.37) is from zero, assuming the null hypothesis is true.
Statisticians compare this Z-score against critical values from the Z-distribution table to decide whether to support or refute the null hypothesis. With an alpha level of 0.01 and a one-tailed test, the critical value is 2.33. Since 1.45 does not exceed 2.33, the null hypothesis remains intact, hinting that the observed difference may not be statistically significant.

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Most popular questions from this chapter

Harper's Index reported that \(80 \%\) of all supermarket prices end in the digit 9 or \(5 .\) Suppose you check a random sample of 115 items in a supermarket and find that 88 have prices that end in 9 or \(5 .\) Does this indicate that less than \(80 \%\) of the prices in the store end in the digits 9 or 5 ? Use \(\alpha=0.05\).

A random sample of \(n_{1}=288\) voters registered in the state of California showed that 141 voted in the last general election. A random sample of \(n_{2}=216\) registered voters in the state of Colorado showed that 125 voted in the most recent general election. (See reference in Problem 25.) Do these data indicate that the population proportion of voter turnout in Colorado is higher than that in California? Use a \(5 \%\) level of significance.

Compare statistical testing with legal methods used in a U.S. court setting. Then discuss the following topics in class or consider the topics on your own. Please write a brief but complete essay in which you answer the following questions. (a) In a court setting, the person charged with a crime is initially considered to be innocent. The claim of innocence is maintained until the jury returns with a decision. Explain how the claim of innocence could be taken to be the null hypothesis. Do we assume that the null hypothesis is true throughout the testing procedure? What would the alternate hypothesis be in a court setting? (b) The court claims that a person is innocent if the evidence against the person is not adequate to find him or her guilty. This does not mean, however, that the court has necessarily proved the person to be innocent. It simply means that the evidence against the person was not adequate for the jury to find him or her guilty. How does this situation compare with a statistical test for which the conclusion is "do not reject" (i.e., accept) the null hypothesis? What would be a type II error in this context? (c) If the evidence against a person is adequate for the jury to find him or her guilty, then the court claims that the person is guilty. Remember, this does not mean that the court has necessarily proved the person to be guilty. It simply means that the evidence against the person was strong enough to find him or her guilty. How does this situation compare with a statistical test for which the conclusion is to "reject" the null hypothesis? What would be a type I error in this context? (d) In a court setting, the final decision as to whether the person charged is innocent or guilty is made at the end of the trial, usually by a jury of impartial people. In hypothesis testing, the final decision to reject or not reject the null hypothesis is made at the end of the test by using information or data from an (impartial) random sample. Discuss these similarities between statistical hypothesis testing and a court setting. (e) We hope that you are able to use this discussion to increase your understanding of statistical testing by comparing it with something that is a well. known part of our American way of life. However, all analogies have weak points. It is important not to take the analogy between statistical hypothesis testing and legal court methods too far. For instance, the judge does not set a level of significance and tell the jury to determine a verdict that is wrong only \(5 \%\) or \(1 \%\) of the time. Discuss some of these weak points in the analogy between the court setting and hypothesis testing.
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