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The director of the Kaiser Family Foundation's Program for the Study of Entertainment Media and Health said, "It's not just teenagers who are wired up and tuned in, its babies in diapers as well." A study by Kaiser Foundation provided one of the first looks at media use among the very youngest children- those from 6 months to 6 years of age (Kaiser Family Foundation. 2003 , www .kff.org). Because previous research indicated that children. who have a TV in their bedroom spend less time reading than other children, the authors of the Foundation study were interested in learning about the proportion of kids who have a TV in their bedroom. They collected data from two independent random samples of parents. One sample consisted of parents of children age 6 months to 3 years old. The second sample consisted of parents of children age 3 to 6 years old. They found that the proportion of children who had a TV in their bedroom was \(.30\) for the sample of children age 6 months to 3 years and \(.43\) for the sample of children age 3 to 6 years old. Suppose that the two sample sizes were each 100 . a. Construct and interpret a \(95 \%\) confidence interval for the proportion of children age 6 months to 3 years who have a TV in their bedroom. Hint: This is a one-sample confidence interval. b. Construct and interpret a \(95 \%\) confidence interval for the proportion of children age 3 to 6 years who have a 'TV in their bedroom. c. Do the confidence intervals from Parts (a) and (b) overlap? What does this suggest about the two population proportions? d. Construct and interpret a \(95 \%\) confidence interval for the difference in the proportion that have TVs in the bedroom for children age 6 months to 3 years and for children age 3 to 6 years. e. Is the interval in Part (d) consistent with your answer in Part (c)? Explain.

Step by step solution

01

Construct 95% Confidence Interval for First Proportion

We will use the formula for a confidence interval for a proportion to solve this. The sample proportion for the group of children aged 6 months to 3 years is 0.30, the sample size is 100 and the z-score for a 95% confidence level is approximately 1.96. So the confidence interval is: \( 0.30 \pm 1.96 \times \sqrt{(0.30(1-0.30)/100)} \). Calculate this to find the confidence interval.

02

Construct 95% Confidence Interval for Second Proportion

Similarly, use the formula for a confidence interval for a proportion as in the previous step. The sample proportion for the group of children aged 3 to 6 years is 0.43, the sample size is again 100 and the z-score for a 95% confidence level is still approximately 1.96. So the confidence interval is: \( 0.43 \pm 1.96 \times \sqrt{(0.43(1-0.43)/100)} \). Calculate this to find the confidence interval.

03

Determine if Confidence Intervals Overlap

To determine if the confidence intervals overlap, compare the lower limit of the upper interval with the upper limit of the lower interval. If the lower limit of the upper interval is greater than the upper limit of the lower interval, then the confidence intervals do not overlap.

04

Construct 95% Confidence Interval for Difference in Proportions

We now construct a 95% confidence interval for the difference in proportions. Use the formula \((p1 - p2) \pm 1.96 \times \sqrt{(p1(1-p1)/n1 + p2(1-p2)/n2)}\). In this case, p1 is 0.30, n1 is 100, p2 is 0.43 and n2 is 100. So the confidence interval is \( (0.30-0.43) \pm 1.96 \times \sqrt{(0.30(1-0.30)/100 + 0.43(1-0.43)/100)} \). Calculate this to find the confidence interval.

05

Compare Interval in Part d with Overlap in Part c

Check if the interval in Part d is consistent with your answer in Part c. If the confidence intervals in Part c overlap, then the interval in Part d should include zero. If the intervals in Part c do not overlap, then the interval in Part d should not include zero.

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Key Concepts

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

Statistical Analysis

Statistical analysis allows researchers to collect, analyze, and interpret data to make informed decisions. It's pivotal in research as it gives meaning to raw data, turning numbers into a story about the population under study. In the context of the exercise from the Kaiser Family Foundation study, statistical analysis is used to determine the proportion of young children with a TV in their bedroom through calculated confidence intervals. By using known formulas and statistical techniques, researchers can infer population characteristics with a specified level of certainty, in this case, 95%.

The essence lies in using measures like sample proportions, sample sizes, and pre-determined confidence levels (such as 95%) to infer about the population's behavior. Moreover, calculating these intervals helps in understanding whether the observed differences between groups are statistically significant or could have occurred by chance. This method reduces uncertainty and supports scientific conclusions with quantifiable evidence.

Proportion

A proportion refers to the part of a whole and is often expressed as a fraction or percentage. In the Kaiser Family Foundation study, the proportions of interest reflect the percentage of children in two different age groups who have a TV in their bedroom. The main purpose of analyzing proportions is to understand how common a particular trait or behavior is within a population or subgroup.

To find the confidence interval for a proportion, like 0.30 for children aged 6 months to 3 years, the formula \( \hat{p} \pm z \times \sqrt{(\hat{p}(1-\hat{p})/n)} \) is used, where \( \hat{p} \) is the sample proportion, \( z \) is the z-score for the confidence level, and \( n \) is the sample size. This calculation gives a range where the true population proportion is likely to lie. For example, a 95% confidence interval implies that we can be 95% confident that the interval contains the true proportion of the population with this characteristic. In practical terms, this method provides a benchmark to assess whether observed differences in characteristics, like having a TV in the bedroom, are statistically widescale or just randomly distributed.

Difference in Proportions

The study not only aimed to estimate the proportion of children with TVs in their bedroom but also to explore if there is a significant difference between the two age groups. This difference in proportions is crucial to understanding if the behavior (having a TV in the bedroom) becomes more prevalent as children grow older.

To calculate the confidence interval for the difference in proportions, we use the formula:\( (p_1 - p_2) \pm z \times \sqrt{(p_1(1-p_1)/n_1 + p_2(1-p_2)/n_2)} \), where \( p_1 \) and \( p_2 \) are the sample proportions, and \( n_1 \) and \( n_2 \) are the sample sizes of the two groups. This interval gives us a range of values within which the true difference of the population proportions is likely to fall.

In this study, if the interval contains zero, it suggests there may be no significant difference between the two age groups. Otherwise, if zero is not included, it indicates a significant difference. This aspect of analysis is important for making educational and policy decisions, as it highlights potentially changing media behavior patterns as children age.