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Chapter 10: Problem 13

Who owns iPods? As part of the Pew Internet and American Life Project, researchers surveyed a random sample of 800 teens and a separate random sample of 400 young adults. For the teens, \(79 \%\) said that they own an iPod or MP3 player. For the young adults, this figure was \(67 \%\). Do the data give convincing evidence of a difference in the proportions of all teens and young adults who would say that they own an iPod or MP3 player? State appropriate hypotheses for a test to answer this question. Define any parameters you use.

Short Answer

Expert verified
The hypotheses test investigates if there is a difference, with null hypothesis \(H_0: p_1 = p_2\) and alternative \(H_a: p_1 \neq p_2\).

Step by step solution

01

Define the Parameters

Let \(p_1\) represent the proportion of all teens who own an iPod or MP3 player and \(p_2\) represent the proportion of all young adults who own an iPod or MP3 player.
02

State the Null Hypothesis

The null hypothesis \(H_0\) is that there is no difference in the proportions, i.e., \(p_1 - p_2 = 0\).
03

State the Alternative Hypothesis

The alternative hypothesis \(H_a\) is that there is a difference in the proportions, i.e., \(p_1 - p_2 eq 0\).
04

Identify Sample Proportions

Calculate the sample proportions: \(\hat{p}_1 = \frac{79}{100} = 0.79\) for teens and \(\hat{p}_2 = \frac{67}{100} = 0.67\) for young adults.
05

Calculate Standard Error

Use the formula for standard error of the difference in proportions: \[ SE = \sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2}} \] where \(n_1 = 800\) and \(n_2 = 400\).
06

Calculate the Test Statistic

Compute the test statistic using: \[ z = \frac{(\hat{p}_1 - \hat{p}_2)}{SE} \] where \(\hat{p}_1 - \hat{p}_2 = 0.79 - 0.67\).
07

Make a Decision

Compare the calculated \(z\)-value with the critical \(z\)-value for a chosen significance level. If \(|z|\) is greater than the critical value, reject \(H_0\).
08

State Conclusion

If \(H_0\) is rejected, conclude that there is convincing evidence of a difference in the proportions. Otherwise, the difference is not statistically significant.

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Key Concepts

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

Proportions Comparison
When comparing proportions, we look at percentages that reflect the characteristics we are interested in, such as ownership of iPods or MP3 players. Here, we want to understand how these ownership rates vary between teens and young adults. Comparing these proportions helps us determine if there is a statistically meaningful difference between two groups.
  • Calculate individual sample proportions. For example, 79% of teens and 67% of young adults own the devices, hence their sample proportions are 0.79 and 0.67 respectively.
  • Determine if these differences are large enough to suggest an actual difference in the populations they represent.
Capturing these comparison points aids in making conclusions about the broader population based on the sample statistics.
Null and Alternative Hypotheses
In hypothesis testing, we use the null and alternative hypotheses to test our assumptions. The null hypothesis (\(H_0\)) represents the idea of no effect or no difference. Here, we state that there is no difference in iPod ownership proportions between teens and young adults.
  • Null Hypothesis (\(H_0\)): \(p_1 - p_2 = 0\). This means the proportion of teens owning iPods equals that of young adults.
  • Alternative Hypothesis (\(H_a\)): \(p_1 - p_2 eq 0\). This implies that the ownership rates differ between the two age groups.
Choosing the correct hypothesis to test affects the validity of our conclusions. Hypothesis testing helps us determine whether observed differences reflect reality or are simply due to chance.
Standard Error Calculation
The standard error quantifies the variability of a sample statistic. In our scenario, it's crucial to calculate the standard error for the difference between proportions. This helps us understand the expected fluctuation of sample proportions due to random sampling.
To calculate the standard error:
  • Use the formula: \[SE = \sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2}}\]
  • Where \(\hat{p}_1\) and \(\hat{p}_2\) are the sample proportions; \(n_1\) and \(n_2\) are the sample sizes.
A lower standard error indicates more reliable estimates, making it easier to determine if the observed difference between the groups is significant.
Statistical Significance
Statistical significance helps us assess whether our findings hold true beyond just the sample data. When comparing proportions, we calculate a test statistic, typically a z-score, to identify the likelihood of observing our data, assuming the null hypothesis is true.
To determine significance:
  • Calculate the z-score using the formula: \[ z = \frac{(\hat{p}_1 - \hat{p}_2)}{SE} \]
  • Compare the calculated z-value against the critical z-value determined by the chosen significance level, often 0.05.
  • If \(|z|\) is greater than the critical value, the result is statistically significant, meaning we have enough evidence to reject the null hypothesis.
Understanding statistical significance ensures that we make informed decisions about the presence of real differences between the populations.

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Most popular questions from this chapter

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