A concern of Major League Baseball is that games last too long.Some executives in the league's headquarters believe that the mean length of games this past year exceeded 3 hours (180 minutes).To test this,the league selected a random sample of 80 games and found the following results: = 193 minutes and s = 16 minutes. Based on these results,if the null hypothesis is tested using an alpha level equal to 0.10,which of the following is true?
A)The null hypothesis should be rejected if 182.31.
B)The test statistic is t = 1.2924.
C)Based on the sample data,the null hypothesis cannot be rejected.
D)It is possible that when the hypothesis test is completed that a Type II statistical error has been made.

Question

A concern of Major League Baseball is that games last too long.Some executives in the league's headquarters believe that the mean length of games this past year exceeded 3 hours (180 minutes).To test this,the league selected a random sample of 80 games and found the following results:  = 193 minutes and s = 16 minutes. Based on these results,if the null hypothesis is tested using an alpha level equal to 0.10,which of the following is true?
A)The null hypothesis should be rejected if > 182.31. 
B)The test statistic is t = 1.2924. 
C)Based on the sample data,the null hypothesis cannot be rejected. 
D)It is possible that when the hypothesis test is completed that a Type II statistical error has been made.

Quizplus · Accepted Answer

We need to test the null hypothesis that the mean length of the games this past year is not greater than 180 minutes.

Step 1: State the null and alternative hypotheses
H0: μ ≤ 180 (the mean length is not greater than 180 minutes)
Ha: μ > 180 (the mean length is greater than 180 minutes)

Step 2: Determine the level of significance
α = 0.10

Step 3: Calculate the test statistic and p-value
t = (x̄ - μ) / (s / sqrt(n))
t = (193 - 180) / (16 / sqrt(80))
t = 4.25

p-value = P(t > 4.25) = 0.00003

Step 4: Make a decision and interpret the results
Since the p-value (0.00003) is less than the level of significance (0.10), we reject the null hypothesis. This means that there is sufficient evidence to conclude that the mean length of the games this past year is greater than 180 minutes.
 
To find the critical value of t with α = 0.10 and 79 degrees of freedom, we can use a t-distribution table or calculator. The critical value is 1.2892. Since our calculated t-value is greater than this critical value, we can conclude that the null hypothesis should be rejected if 193 > 182.31. Therefore, choice A is correct.

There is no mention of a Type II statistical error in the question, so choice D is not applicable. Choice B is incorrect because the calculated t-value is not 1.2924. Choice C is incorrect because we have just rejected the null hypothesis based on the given sample data.

A Concern of Major League Baseball Is That Games Last