91Ó°ÊÓ

TV watching A researcher predicts that the percentage of people who do not watch TV is higher now than before the advent of the Internet. Let \(p_{1}\) denote the population proportion of American adults in 1975 who reported watching no TV. Let \(p_{2}\) denote the corresponding population proportion in 2008 . a. Set up null and alternative hypotheses to test the researcher's prediction. b. According to General Social Surveys, 57 of the 1483 subjects sampled in 1975 and 87 of the 1324 subjects sampled in 2008 reported watching no TV. Find the sample estimates of \(p_{1}\) and \(p_{2}\). c. Show steps of a significance test. Explain whether the results support the reseacher's claim.

Step by step solution

01

State Hypotheses

For hypothesis testing, we establish the null hypothesis \(H_0: p_1 \geq p_2\), indicating the proportion in 1975 was at least equal to that in 2008. The alternative hypothesis is \(H_a: p_1 < p_2\), reflecting the researcher's claim that the proportion of non-TV watchers has increased.

02

Calculate Sample Proportions

Calculate the sample proportion for 1975: \( \hat{p}_1 = \frac{57}{1483} \approx 0.0384 \) and for 2008: \( \hat{p}_2 = \frac{87}{1324} \approx 0.0657 \). These represent the estimated proportions of non-TV watchers in each year.

03

Compute Test Statistic

We use the formula for the test statistic in a proportion comparison: \[ z = \frac{\hat{p}_1 - \hat{p}_2}{\sqrt{\hat{p}(1-\hat{p})(\frac{1}{n_1} + \frac{1}{n_2})}} \]Where \(\hat{p} = \frac{x_1 + x_2}{n_1 + n_2} = \frac{57 + 87}{1483 + 1324} \approx 0.0515\). Calculate the z-score to evaluate the significance.

04

Find Critical Value and Compare

Under the significance level (typically 0.05), find the critical z-value from the standard normal distribution table. If the calculated z-value is less than this critical value, the null hypothesis is rejected in favor of the alternative hypothesis.

05

Conclusion of Hypothesis Test

Compare the calculated z-value with the critical value. If the z-value falls into the critical region, reject \(H_0\). A rejection supports the researcher's claim that the proportion of non-TV watchers has increased. Otherwise, fail to reject \(H_0\).

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

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

Significance Test

Performing a significance test allows us to determine whether a hypothesis about a population parameter is supported by sample data. This process helps us quantify the evidence against the null hypothesis. Here, we look at the difference between two population proportions over two different time periods, given by sample data in 1975 and 2008.
The steps generally involve:

• Stating the null and alternative hypotheses.
• Calculating the test-statistic using sample data.
• Choosing an appropriate significance level (\( \alpha \)) to determine the critical value or region for decision making.
• Interpreting the results to make a conclusion.

For this exercise, a z-test is used because it is suited for comparing the proportions from independent samples. The z-test for proportions helps determine if there is a significant difference between the proportions of non-TV watchers in the two considered years.

Sample Proportion

The sample proportion is a vital element in hypothesis testing, especially when dealing with proportions. It represents the fraction of the sample that exhibits the characteristic of interest, in this case, the people not watching TV.
In the context of our problem:

• The sample proportion for 1975, noted as \( \hat{p}_1 \), is calculated as\[ \hat{p}_1 = \frac{57}{1483} \approx 0.0384 \]meaning about 3.84% of the sampled 1975 population did not watch TV.
• For 2008, the sample proportion, noted as \( \hat{p}_2 \), is:\[ \hat{p}_2 = \frac{87}{1324} \approx 0.0657 \]indicating about 6.57% of the 2008 sample were non-TV watchers.

These sample proportions give us an estimate to help compare the trends over time and support our hypothesis testing.

Null and Alternative Hypotheses

Setting up null and alternative hypotheses is the first step in hypothesis testing. These hypotheses represent statements about population parameters that we test using sample data.
In this example, we are comparing the proportions of non-TV watchers between two separate years.

• The **null hypothesis**, denoted \( H_0 \), assumes that there is no increase in the proportion, stating: \( H_0: p_1 \geq p_2 \)where \( p_1 \) is the proportion of 1975 non-TV watchers and\( p_2 \) is 2008's.
• The **alternative hypothesis** represents the researcher’s claim, suggesting an increase, stated as: \( H_a: p_1 < p_2 \)implying the 2008 proportion is indeed greater.

These hypotheses form the basis for our statistical test and guide us on determining whether the change over years is statistically significant or if it merely occurred by random chance.