Ch. 7 - Estimating Parameters and Determining Sample Sizes

Triola - Elementary Statistics 14th Edition

Triola14th EditionElementary StatisticsISBN: 9780137366446Not the one you use?Change textbook

Triola 14th Edition

Ch. 7 - Estimating Parameters and Determining Sample Sizes

Problem 7.r.9a

Chapter 7, Problem 7.r.9a

Alcohol in Children’s Movies Listed below is a simple random sample of times (seconds) that animated children’s movies showed the use of alcohol (based on Data Set 20 “Alcohol and Tobacco in Movies” in Appendix B).

a. Are the requirements for constructing a 95% confidence interval estimate of the population mean satisfied? If so, construct that confidence interval.

Verified step by step guidance

Step 1: Verify the requirements for constructing a 95% confidence interval. The requirements are: (a) the sample is a simple random sample, (b) the population is normally distributed or the sample size is large (n > 30). In this case, the sample size is 30, which is borderline, so we can proceed assuming the Central Limit Theorem applies.

Step 2: Calculate the sample mean (\( \bar{x} \)) using the formula \( \bar{x} = \frac{\sum x_i}{n} \), where \( x_i \) are the individual data points and \( n \) is the sample size.

Step 3: Calculate the sample standard deviation (\( s \)) using the formula \( s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}} \). This measures the spread of the data around the mean.

Step 4: Determine the margin of error (E) using the formula \( E = t_{\alpha/2} \cdot \frac{s}{\sqrt{n}} \), where \( t_{\alpha/2} \) is the critical t-value for a 95% confidence level and \( n-1 \) degrees of freedom. Use a t-distribution table or calculator to find \( t_{\alpha/2} \).

Step 5: Construct the confidence interval using the formula \( \bar{x} \pm E \). This gives the range of values within which the population mean is likely to fall with 95% confidence.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Simple Random Sample

A simple random sample is a subset of individuals chosen from a larger population, where each individual has an equal chance of being selected. This method ensures that the sample is representative of the population, minimizing bias and allowing for valid statistical inferences. In the context of the question, the sample of times showing alcohol use in children's movies is assumed to be randomly selected to accurately reflect the broader population of such movies.

Confidence Interval

A confidence interval is a range of values, derived from a sample, that is likely to contain the population parameter (such as the mean) with a specified level of confidence, typically 95%. It provides an estimate of uncertainty around the sample mean, indicating how much the sample mean might vary from the true population mean. Constructing a confidence interval involves calculating the sample mean, standard deviation, and using the appropriate critical value from the t-distribution or z-distribution.

Normality Assumption

The normality assumption states that for many statistical methods, including the construction of confidence intervals, the data should be approximately normally distributed, especially as sample sizes increase. This assumption is crucial when determining the validity of the confidence interval. If the sample size is small, normality can be assessed using graphical methods or tests, while larger samples can rely on the Central Limit Theorem, which states that the sampling distribution of the mean will be approximately normal regardless of the population distribution.

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