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The article "The Assodation Between Television Viewing and Irregular Sleep Schedules Among Children Less Than 3 Years of Age" (Pediatrics 120051 : \(851-856\) ) reported the accompanying \(95 \%\) confidence intervals for average TV viewing time (in hours per day) for three different age groups. Confidence Interval Age Group \(=\) \(95 \%\) Less than (0.8,1.0) 12 12 to 23 2 months \begin{tabular}{l} (0.8,1.0) \\ (1.4,1.8) \\ (2.1,2.5) \\ \hline \end{tabular} \begin{tabular}{r} 12 to 2 \\ 24 to 3 \\ \hline \end{tabular} n 12 montns 5 months months 35 a. Suppose that the sample sizes for each of the three age group samples were equal. Based on the given confidence intervals, which of the age group samples had the greatest variability in TV viewing time? Explain your choice. b. Now suppose that the sample standard deviations for the three age group samples were equal, but that the three sample sizes might have been different. Which of the three age-group samples had the largest sample size? Explain your choice. c. The interval (.768,1.032) is either a \(90 \%\) confidence interval or a \(99 \%\) confidence interval for the mean TV viewing time computed using the sample data for children less than 12 months old. Is the confidence level for this interval \(90 \%\) or \(99 \%\) ? Explain your choice.

Step by step solution

01

Identifying variability

The age group sample with the greatest variability in TV viewing time would be the one with the widest confidence interval. This is because the width of a confidence interval reflects the variability of the data. Here, the age group '24 to 35 months' has the largest confidence interval (2.1, 2.5), hence it has the greatest variability.

02

Inferring sample sizes

If the sample standard deviations for the three age group samples were equal but the sample sizes might have been different, the age group sample with the smallest confidence interval would have the largest sample size. This is because larger sample sizes provide more precise estimates, thus resulting in narrower confidence intervals. In this case, the age group 'Less than 12 months' has the smallest confidence interval (0.8, 1.0), hence it has the largest sample size.

03

Determining the confidence level

The interval (0.768,1.032) is narrower than the 95% confidence interval (0.8,1.0) for the same age group 'Less Than 12 months'. Therefore this would be a 90% confidence level as lower confidence levels result in narrower confidence intervals given the same sample data.

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Key Concepts

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

Confidence Interval

A confidence interval is a range used to estimate the true value of a population parameter, like the mean. It is expressed with two numbers: the lower and upper bounds. For example, the report provides a 95% confidence interval for the average TV viewing time for kids between zero and twelve months as (0.8, 1.0) hours per day.

What this means is that we can be 95% confident that the true average TV viewing time for the whole population of children in this age range is between 0.8 and 1.0 hours. A key feature of the confidence interval is its width. A wider interval suggests more variability in the data or a smaller sample size. A narrower interval indicates less variability or a larger sample size.

Confidence levels, such as 90%, 95%, or 99%, express how often the method, used to generate the interval, includes the true parameter value across many samples. Lower confidence levels yield narrower intervals but less certainty, while higher levels provide wider intervals with more certainty.

When researchers report confidence intervals in studies, they provide valuable information about the precision and reliability of their estimates.

Sample Size

The sample size, often denoted by "n," refers to the number of observations in a sample. It plays a crucial role in determining the confidence interval's width. A larger sample size generally results in a more accurate and narrower confidence interval.

In the exercise, if the standard deviations are the same for all age groups, the group with the smallest interval (Less than 12 months) implies it was derived from the largest sample size. This happens because with more data points, the estimate becomes more precise, shrinking the interval.

A larger sample provides better insight into the true population mean, hence offering more reliable results. In practice, researchers aim to choose an adequate sample size to balance the logistical constraints (such as time and cost) with the need for precise estimates.

Standard Deviation

The standard deviation is a measure of how spread out the numbers in a data set are. It shows the extent of variation or dispersion of the set of values. A small standard deviation means that the values tend to be close to the mean of the set, while a large standard deviation indicates that the values are spread out over a wider range.

In the given problem, if we assume equal sample sizes, different standard deviations would create the varying widths of the confidence intervals. For example, the group '24 to 35 months' had the widest interval (2.1, 2.5), suggesting more variability in their TV watching times, and thus, a higher standard deviation.

Understanding standard deviation is vital as it impacts the reliability of our confidence intervals. Higher dispersion (standard deviation) in the data generally requires a larger sample size to achieve a similar precision in the confidence interval as a data set with lower dispersion.