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Chapter 10: Problem 15

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%. Is there a significant difference between the population proportions? State appropriate hypotheses for a significance test to answer this question. Define any parameters you use.

Short Answer

Expert verified
There is a significant difference between the proportions of teens and young adults who own iPods or MP3 players.

Step by step solution

01

Define Parameters

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

State Hypotheses

We are testing whether there is a significant difference between \( p_1 \) and \( p_2 \). The null hypothesis \( H_0 \) is that there is no difference in the population proportions, i.e., \( p_1 = p_2 \). The alternative hypothesis \( H_a \) is that there is a difference, i.e., \( p_1 eq p_2 \).
03

Calculate Sample Proportions

The sample proportion for teens, \( \hat{p_1} = \frac{79}{100} = 0.79 \). The sample proportion for young adults, \( \hat{p_2} = \frac{67}{100} = 0.67 \).
04

Determine Pooled Proportion

The pooled proportion \( \hat{p} \) is calculated as follows: \( \hat{p} = \frac{(0.79 \times 800) + (0.67 \times 400)}{800 + 400} = 0.75 \).
05

Compute Standard Error

The standard error (SE) of the difference in proportions is given by \( SE = \sqrt{ \hat{p}(1-\hat{p}) \left( \frac{1}{n_1} + \frac{1}{n_2} \right) } \). Substituting the values, \( SE = \sqrt{0.75 \times 0.25 \times \left( \frac{1}{800} + \frac{1}{400} \right)} = 0.0272 \).
06

Calculate Test Statistic

The test statistic \( z \) for comparing the two proportions is given by \( z = \frac{\hat{p_1} - \hat{p_2}}{SE} = \frac{0.79 - 0.67}{0.0272} = 4.41 \).
07

Make a Conclusion

Since the calculated \( z \) value of 4.41 is greater than the critical value of 1.96 for a 95% confidence level, we reject the null hypothesis. Thus, there is a significant difference between the proportion of teens and young adults who own iPods or MP3 players.

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Key Concepts

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

Population Proportions
Population proportions are the fractions of a population exhibiting a certain characteristic. In this exercise, we're interested in the proportion of teens and young adults who own an iPod or MP3 player.
For teens, the population proportion is represented by \( p_1 \), while for young adults, it鈥檚 \( p_2 \).
Understanding these proportions helps in determining how common it is for individuals in these age groups to own such devices.
The key is to compare these proportions from the samples to see if they reflect real differences in the broader populations. This comparison is useful in surveys and market research to understand trends and preferences.
Significance Test
A significance test is a statistical method used to determine whether the observed differences between groups (in this case, population proportions) are genuine or could have occurred by random chance.
In our scenario, we are testing the claim about the difference between teen and young adult ownership of iPods or MP3 players.
  • The null hypothesis \( H_0 \) posits that no difference exists: \( p_1 = p_2 \).
  • The alternative hypothesis \( H_a \) suggests a difference does occur: \( p_1 eq p_2 \).
This process enables us to make informed conclusions about the wider population using sample data.
Standard Error
Standard error (SE) quantifies the amount of variation in a sampling distribution. It's crucial for evaluating how sample data compares to the true population.
Specifically, when comparing two proportions, the standard error helps us understand the extent to which the observed difference might deviate from the true population difference by chance.
Calculation of SE uses the pooled proportion \( \hat{p} \) and both sample sizes:
  • The formula is: \( SE = \sqrt{ \hat{p}(1-\hat{p}) \left( \frac{1}{n_1} + \frac{1}{n_2} \right) } \).
  • This equation considers both sample groups and their respective sizes.
This measure ensures that our test statistic accurately reflects any real differences in the sample data.
Test Statistic
The test statistic is a calculated value used to determine the significance of the test. It converts the observed difference between sample proportions into a measure that can be compared to a known distribution.
In this case, the test statistic is Z, calculated using:
  • The formula: \( z = \frac{\hat{p_1} - \hat{p_2}}{SE} \).
  • This determines how many standard errors the observed difference \( \hat{p_1} - \hat{p_2} \) is away from zero.
A higher absolute value of \( z \) indicates a lesser likelihood that the observed difference is due to random chance alone.
In our example, a \( z \)-value greater than the critical value leads us to reject the null hypothesis, concluding there's a significant difference between the groups.

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