Question

Do you own an iPod Nano or a Sony Walkman Bean? These and other brands of MP3 players are becoming more and more popular among younger Americans. An iPod survey reported that \(54 \%\) of 12 - to 17 -year-olds, \(30 \%\) of 18 - to 34 -year-olds, and \(13 \%\) of 35 - to 54 -year-olds own MP3 players. \({ }^{6}\) Suppose that these three estimates are based on random samples of size \(400,350,\) and \(362,\) respectively. a. Construct a \(95 \%\) confidence interval estimate for the proportion of 12 - to 17 -year-olds who own an MP3 player. b. Construct a \(95 \%\) confidence interval estimate for the proportion of 18 - to 34 -year-olds who own an MP3 player.

Short answer

Question: Determine the 95% confidence intervals for the proportion of 12-to-17-year-olds and 18-to-34-year-olds who own an MP3 player. Answer: The 95% confidence interval estimate for the proportion of 12-to-17-year-olds who own an MP3 player is (0.48994, 0.59006), and the 95% confidence interval estimate for the proportion of 18-to-34-year-olds who own an MP3 player is (0.25153, 0.34847).

Step-by-step solution

Calculate the sample proportions for each age group

Given the survey results, we can calculate the sample proportions of MP3 player ownership for each age group: 12-to-17-year-olds: \(\hat{p}_1 = 0.54\) 18-to-34-year-olds: \(\hat{p}_2 = 0.30\)

Determine the critical value corresponding to 95% confidence

For a 95% confidence interval, the critical value \(Z_{\frac{\alpha}{2}}\) can be found in a Z-table or using a calculator. The critical value is: \(Z_{\frac{\alpha}{2}} = 1.96\)

Calculate the confidence interval for 12-to-17-year-olds

Using the formula for the confidence interval, we can calculate the 95% confidence interval for the proportion of 12-to-17-year-olds who own an MP3 player: \(CI_1 = \hat{p}_1 \pm Z_{\frac{\alpha}{2}} \times \sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1}}\) \(CI_1 = 0.54 \pm 1.96 \times \sqrt{\frac{0.54(1-0.54)}{400}}\) \(CI_1 = 0.54 \pm 1.96 \times \sqrt{0.0006475}\) \(CI_1 = 0.54 \pm 0.05006\) Therefore, the 95% confidence interval estimate for the proportion of 12-to-17-year-olds who own an MP3 player is (0.48994, 0.59006).

Calculate the confidence interval for 18-to-34-year-olds

Now, we can calculate the 95% confidence interval for the proportion of 18-to-34-year-olds who own an MP3 player: \(CI_2 = \hat{p}_2 \pm Z_{\frac{\alpha}{2}} \times \sqrt{\frac{\hat{p}_2(1-\hat{p}_2)}{n_2}}\) \(CI_2 = 0.30 \pm 1.96 \times \sqrt{\frac{0.30(1-0.30)}{350}}\) \(CI_2 = 0.30 \pm 1.96 \times \sqrt{0.00061286}\) \(CI_2 = 0.30 \pm 0.04847\) Therefore, the 95% confidence interval estimate for the proportion of 18-to-34-year-olds who own an MP3 player is (0.25153, 0.34847).

Key concepts

Sample Proportion

In statistics, the sample proportion is the percentage of individuals in a sample that have a particular characteristic. For example, in a survey, it measures the fraction of people who own an MP3 player in different age groups. Calculating sample proportion, denoted as \( \hat{p} \), is straightforward. You divide the number of favorable responses by the total number of sample responses.

• For the 12-to-17-year-olds, if 54 out of 100 surveyed own an MP3 player, the sample proportion \( \hat{p}_1 \) is 0.54.
• For the 18-to-34-year-olds, with a proportion of 30%, \( \hat{p}_2 \) is 0.30.

Sample proportions help gauge the prevalence of characteristics in a population. They provide a groundwork for estimating and creating confidence intervals. With a bigger sample size, the sample proportion's estimate is more likely to reflect the true proportion in the whole population.

Critical Value

The critical value is an important factor in hypothesis testing and constructing confidence intervals. It represents the cutoff point that determines the extent of uncertainty in a sample statistic based on a desired level of confidence. This value is derived from the Z-distribution or standard normal distribution when dealing with proportions.
For a 95% confidence interval, which is common for social sciences, the critical value \( Z_{\frac{\alpha}{2}} \) is 1.96.

• This means that if you want to be 95% certain your confidence interval contains the true population proportion, you include 1.96 standard deviations (to the left and right) of your sample proportion.

The critical value comes into play when calculating the margin of error in a confidence interval: the larger the critical value, the wider the interval, reflecting the inherent uncertainty.

Z-score

A Z-score measures exactly how many standard deviations a data point is from the mean. In calculating the confidence interval for a sample proportion, the Z-score is used to assess the probability that a sample mean will fall within a certain range of the population mean.
The Z-score is crucial when it comes to proportions, as the sample mean of the proportion under the null hypothesis is normally distributed, allowing use of the normal Z-distribution.

• This standardization helps in determining the position of data points in relation to the overall distribution.

Using the Z-score helps in creating confidence intervals by specifying the range of proportion values that are statistically not different from the population proportion.

MP3 Player Ownership Statistics

Understanding MP3 player ownership statistics shines a light on consumer habits across various demographics. It helps us gauge the penetration and popularity of such technology over different age groups. Take iPods or Sony Walkman Beans, for example. Ownership statistics reveal how mainstream these gadgets have become.

• Among 12-to-17-year-olds, 54% own an MP3 player, indicating strong popularity in this group.
• The 18-to-34-year-olds have a 30% ownership rate, showing steady, though less intense, engagement.
• Only 13% of 35-to-54-year-olds own one, illustrating slower adoption in older age groups.

By analyzing these statistics, marketers and manufacturers can target age-specific needs, support product development strategies, and predict market trends. Combining this data with confidence intervals allows a more precise understanding of the needs and size of these market segments.