During the Rose Bowl, the length (in seconds) of 12 randomly...

During the Rose Bowl, the length (in seconds) of 12 randomly chosen commercial breaks during timeouts (following touchdown, turnover, field goal, or punt) were $$ \begin{array}{llllllllllll} 65 & 75 & 85 & 95 & 80 & 100 & 90 & 80 & 85 & 85 & 60 & 65 \end{array} $$ (a) Assuming a normal population, construct a 90 percent confidence interval for the mean length of a commercial break during the Rose Bowl. (b) What are the limitations on your estimate? How could they be overcome?

During the Rose Bowl, the length (in seconds) of 12 randomly chosen commercial breaks during timeouts (following touchdown, turnover, field goal, or punt) were
$$
\begin{array}{llllllllllll}
65 & 75 & 85 & 95 & 80 & 100 & 90 & 80 & 85 & 85 & 60 & 65
\end{array}
$$
(a) Assuming a normal population, construct a 90 percent confidence interval for the mean length of a commercial break during the Rose Bowl. (b) What are the limitations on your estimate? How could they be overcome?

Key Concepts

Confidence Interval

A confidence interval is a statistical tool used to estimate the true value of a population parameter by providing a range based on sample data and a confidence level. It reflects the uncertainty inherent in any sample-based estimate and communicates the degree of precision one can expect from the estimate of the population mean.

t-Distribution

The t-distribution is used for constructing confidence intervals for the population mean when the sample size is small and the population variance is unknown. Unlike the normal distribution, the t-distribution has heavier tails, which means it provides wider intervals to account for the added uncertainty in the estimate.

Normality Assumption

Assuming that the population data follow a normal distribution is crucial when using small sample sizes; it justifies the use of the t-distribution for inference. This assumption ensures that the estimates and associated confidence intervals are reliable and that the statistical methods applied are valid.

Sampling Variability

Sampling variability refers to the natural differences in sample statistics when different random samples are drawn from the same population. Recognizing this concept is essential in understanding that a single sample mean might differ from the true population mean due to random chance, thus motivating the use of interval estimates.

Limitations of the Estimate

Limitations in statistical estimates can arise from a small sample size, potential biases in how the sample is collected, and the reliance on the normality assumption. Such limitations can lead to less reliable or less generalizable results. Addressing these issues often involves increasing the sample size, ensuring that the sampling method is random and representative, and verifying that assumptions about the underlying population distribution hold true.

Please give Ace some feedback