During recent seasons, Major League Baseball has been...

During recent seasons, Major League Baseball has been criticized for the length of the games. A report indicated that the average game lasts 3 hours and 30 minutes. A sample of 17 games revealed the following times to completion. (Note that the minutes have been changed to fractions of hours, so that a game that lasted 2 hours and 24 minutes is reported as 2.40 hours.) $$\begin{array}{|lllllllll|}\hline 2.98 & 2.40 & 2.70 & 2.25 & 3.23 & 3.17 & 2.93 & 3.18 & 2.80 \\2.38 & 3.75 & 3.20 & 3.27 & 2.52 & 2.58 & 4.45 & 2.45 & \\\hline\end{array}$$ Can we conclude that the mean time for a game is less than 3.50 hours? Use the .05 significance level.

During recent seasons, Major League Baseball has been criticized for the length of the games. A report indicated that the average game lasts 3 hours and 30 minutes. A sample of 17 games revealed the following times to completion. (Note that the minutes have been changed to fractions of hours, so that a game that lasted 2 hours and 24 minutes is reported as 2.40 hours.)
$$\begin{array}{|lllllllll|}\hline 2.98 & 2.40 & 2.70 & 2.25 & 3.23 & 3.17 & 2.93 & 3.18 & 2.80 \\2.38 & 3.75 & 3.20 & 3.27 & 2.52 & 2.58 & 4.45 & 2.45 & \\\hline\end{array}$$
Can we conclude that the mean time for a game is less than 3.50 hours? Use the .05 significance level.

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about a population parameter based on sample data. It involves formulating a null hypothesis, which represents a default claim or status quo, and an alternative hypothesis, which reflects the claim to be tested. The process uses sample statistics to determine whether there is enough evidence to reject the null hypothesis in favor of the alternative hypothesis.

Null and Alternative Hypotheses

In the context of statistical testing, the null hypothesis (H0) typically states that there is no effect or no difference, while the alternative hypothesis (H1) posits that there is an effect or a directional difference. For tests aimed at assessing whether a parameter is less than, greater than, or not equal to a specified value, clear specification of these hypotheses is crucial to correctly conducting the test and interpreting the results.

One-sample t-Test

A one?sample t-test is used to test whether the mean of a single sample differs significantly from a known or hypothesized population mean when the population standard deviation is unknown and the sample size is relatively small. This method relies on the t-distribution, which accounts for additional uncertainty due to estimating the population variance using the sample variance.

Significance Level

The significance level, often denoted by alpha (?), is the threshold set by the researcher for rejecting the null hypothesis. It represents the probability of making a Type I error, which is rejecting a true null hypothesis. In hypothesis testing, if the p-value is less than the significance level, the null hypothesis is rejected in favor of the alternative.

p-Value and Test Statistic

The p-value is the probability, under the assumption that the null hypothesis is true, of obtaining a test statistic at least as extreme as the one computed from the sample data. The test statistic itself is a standardized value calculated from the sample data that measures how far the sample mean is from the hypothesized population mean. Comparing the p-value to the significance level helps determine whether the observed difference is statistically significant.

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