Question

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.

Short answer

The estimated proportions of children having a TV in their bedroom for the age groups of 6 months to 3 years and 3 to 6 years are between 21.17% and 38.83% and 34% and 52% respectively. Despite the overlapping of these intervals, the confidence interval for the difference in proportions lies between 0.92% and 25.08%. This suggests that the proportion of children having a TV in the bedroom could be statistically similar between the two age groups.

Step-by-step solution

Confidence Interval of First Group

Using the formula for the confidence interval, for the first group of children from 6 months to 3 years, we have p = 0.30 and n = 100, and Z approximately equals 1.96 for a 95% confidence interval. Substituting these values into the formula gives the confidence interval as \(CI = 0.30 ± 1.96 \sqrt{(0.30(1-0.30))/100} = 0.30 ± 0.0883 = [0.2117, 0.3883]\). Therefore the proportion of children in this age group that have TVs in their bedroom is estimated to be between 21.17% and 38.83%.

Confidence Interval of Second Group

Similarly, for the second group of children from 3 to 6 years old, we have p = 0.43 and n = 100. The confidence interval is \( CI = 0.43 ± 1.96 \sqrt{(0.43(1-0.43))/100} = 0.43 ± 0.09 = [0.34, 0.52]\). Therefore, the proportion of children in this age group who have TVs in their bedroom is estimated to be between 34% and 52%.

Compare Confidence Intervals

The confidence intervals for the two groups are [0.2117, 0.3883] and [0.34, 0.52]. These intervals do overlap, which suggests that the proportions of children having TVs in bedroom for the two age groups could be statistically similar.

Confidence interval for the difference

The difference in the sample proportions is \(0.43 - 0.30 = 0.13\), and the standard error for the difference is computed as \(\sqrt{(0.30(1-0.30)/100)+(0.43(1-0.43)/100)}\). The 95% confidence interval for the difference in the proportions that have TVs in their bedroom for the two age groups is \(0.13 ± 1.96 \times 0.07 = [0.0092, 0.2508]\). It implies that the difference in proportions is estimated to fall between 0.92% and 25.08%.

Consistency of intervals

The interval from Part (d) includes 0 which is consistent with the sliding of parts (a) and (b), implying the proportions of children having a TV in their room between the two age groups could be statistically similar.

Key concepts

Proportion of Children with TV in Bedroom

Understanding the proportion of children who have a TV in their bedroom provides insights into media consumption habits at an early age. In the given exercise, researchers were interested in determining this proportion among two age groups: children aged 6 months to 3 years, and those aged 3 to 6 years. By analyzing these groups separately, we glean how the prevalence of bedroom TVs might change as children grow older. For the younger group, 30% of children were reported to have a TV in their room, whereas 43% of the older group had one. These figures allow us to evaluate media exposure at different developmental stages.
The presence of a TV in a child's bedroom could influence their reading habits and screen time exposure. Parents and educators should be aware of these patterns to make informed decisions about children’s media environments.

Statistical Analysis

Statistical analysis enables us to draw meaningful interpretations from raw data. In this context, it helps assess how common it is for children to have a TV in their bedroom across different age groups. Researchers use confidence intervals to express the range within which we can expect the true proportion to lie, with a given level of certainty, such as 95% in this case. Calculating a confidence interval involves understanding sample proportions, standard error, and using a normal distribution to estimate a range.
The calculated confidence intervals for the two age groups were [0.2117, 0.3883] for children aged 6 months to 3 years and [0.34, 0.52] for those aged 3 to 6 years. These intervals show the expected range for the real proportion of children with TVs in their rooms, taking into account the sampled data.

Sample Size Effect

Sample size plays a crucial role in statistical analysis and determining confidence intervals. It refers to the number of observations used to infer about the population. In the exercise, both age groups had a sample size of 100, which moderately affects the precision and accuracy of the confidence interval.
With a larger sample size, we would achieve narrower confidence intervals, indicating more precise estimations of the true population proportion. A smaller sample size would result in wider intervals due to increased variability, meaning less certainty about the estimate. By choosing 100 as the sample size in each group, researchers strike a balance between logistical feasibility and statistical reliability.

Media Consumption by Children

Children’s media consumption is a trending subject given the steady rise in screen time, starting from a young age. TVs in bedrooms can contribute to increased media consumption. Understanding these trends is vital, as early exposure to media content can affect children's development and daily routines.
In this study, children as young as 6 months were observed, highlighting the early onset of media interaction. As children enter the 3 to 6 years age group, an increase in the percentage of those with TVs in their bedrooms could correlate with changing media habits and preferences. Awareness of these patterns can guide parents and policymakers in making informed decisions about media exposure and its regulatory frameworks.