91Ó°ÊÓ

The article "The Association Between Television Viewing and Irregular Sleep Schedules Among Children Less Than 3 Years of Age" (Pediatrics [2005]: \(851-856\) ) reported the accompanying \(95 \%\) confidence intervals for average TV viewing time (in hours per day) for three different age groups. $$\begin{array}{lc} \text { Age Group } & 95 \% \text { Confidence Interval } \\ \hline \text { Less than } 12 \text { months } & (0.8,1.0) \\ 12 \text { to } 23 \text { months } & (1.4,1.8) \\ 24 \text { to } 35 \text { months } & (2.1,2.5) \\ \hline \end{array}$$ 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

Variability determination

Based on the given confidence intervals, the age group with greatest variability in TV viewing time would be the one with the widest interval. This is due to the fact that a wider confidence interval suggests a higher variability within the sample data. In this case, the age group '24 - 35 months' with the interval \( (2.1,2.5) \) has the greatest variability.

02

Sample size determination

If the sample standard deviations for all age groups are equal, the sample size would be inversely proportional to the width of the confidence interval. This means that the age group with the smallest interval has the largest sample size. So 'Less than 12 months' age group with the interval \( (0.8,1.0) \) has the largest sample size.

03

Confidence level determination

If the interval \( (.768,1.032) \) is a \( 90\% \) confidence interval, it would be narrower than the \( 95\% \) confidence interval \( (0.8,1.0) \) for the same age group 'Less than 12 months'. Conversely, if it is a \( 99\% \) confidence interval, it's expected to be wider. Comparing to the \( 95\% \) confidence interval, the interval \( (.768,1.032) \) is slightly wider. Therefore, it would be a \( 99\% \) confidence interval.

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Key Concepts

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

Variability

When we talk about variability, we are essentially discussing how spread out or consistent our data points are. In the context of confidence intervals, variability can be gauged by the width of these intervals. A wider interval suggests that there is more variability or "spread" within the data.

In the exercise, the age group '24 - 35 months' has the interval (2.1, 2.5). This interval is wider compared to the intervals of the other age groups. A wider interval implies that the individual data points in this group vary quite a bit from one to another.

This is because with more variability, the confidence interval must widen to be sure that it captures the true mean with a given level of confidence, usually set at 95%. Thus, this age group displays the greatest variability in daily TV viewing time.

Sample Size

The concept of sample size plays a crucial role in determining how precise our confidence intervals are. A larger sample size will generally lead to a more precise estimate of the population parameter, resulting in a narrower confidence interval. This is because a larger sample size tends to reduce the error in estimating the population mean.

In the given task, assuming that the variability or standard deviation across the age groups is the same, the sample size will inversely affect the width of the confidence interval. Therefore, the narrower the confidence interval, the larger the sample size.

According to the problem, the 'Less than 12 months' age group has a confidence interval of (0.8, 1.0). It's the narrowest interval among the three groups, which hints that its sample size is the largest. Larger samples provide more information, hence, allowing us to be more precise about where the population mean actually lies.

Confidence Level

Confidence level represents the degree of certainty or probability that the true parameter lies within the calculated confidence interval. It is a reflection of how flexible we are in making errors.

A common misunderstanding is between the levels of confidence, such as 90% and 99%. A higher confidence level, say 99%, would usually result in a wider interval compared to a lower level like 90%. This is because to be more certain (99% sure rather than 90%), we need to widen the range of values we accept as plausible for the parameter.

In the exercise, we are given the interval (0.768, 1.032) which has to be matched with a confidence level. Since it is slightly wider than the existing 95% confidence interval of (0.8, 1.0) for the same age group, it implies that it must be calculated using a higher confidence level of 99%. This wider scope indicates more certainty required in capturing the true mean of TV viewing time.