AnswersCA-Statistics and Probability BUsing the Central Limit TheoremQuestion 3
3
QuizMultiple Choice

Using the Central Limit Theorem

Question 3 • CA-Statistics and Probability B

The times a musician spends performing a rock song and a country song are approximately Normally distributed. The rock song has a mean time of 133 seconds with a standard deviation of 3.5 seconds, and the country song has a mean time of 126 seconds with a standard deviation of 3.3 seconds. If the musician randomly selects 4 times the rock song is played and 3 times the country song is played, what is the probability that the mean time for the rock song is less than the mean time for the country song?

Answer
A
0.0034
B
0.0365
C
0.9964
D
0.9966

Explanation

The difference between the two sample means (rock minus country) is Normally distributed with mean 133 − 126 = 7 seconds and standard deviation √(3.5²/4 + 3.3²/3) ≈ 2.59 seconds. A rock-song mean below the country-song mean means this difference is less than 0, which has z-score (0 − 7)/2.59 ≈ −2.70. The probability is about 0.0034, so the correct choice is 0.0034.

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