To solve these problems, we will use the properties of the normal distribution and the concept of z-scores. A z-score measures how many standard deviations an element is from the mean. The formula to calculate a z-score is: Z = (X - μ) / σ where: X is the value from the dataset, μ is the mean of the dataset, σ is the standard deviation of the dataset. a) More than 20 min? First, we calculate the z-score for 20 minutes: Z = (20 - 15) / 3.5 = 1.43 We then look up this z-score in the z-table, which gives us the probability that a value is less than 20 minutes. The value for 1.43 in the z-table is 0.9236. However, we want the probability that a listener will tune in for more than 20 minutes, so we subtract the value from 1: P(X 20) = 1 - P(X

Question

To solve these problems, we will use the properties of the normal distribution and the concept of z-scores. A z-score measures how many standard deviations an element is from the mean. The formula to calculate a z-score is:

Z = (X - μ) / σ

where:
X is the value from the dataset,
μ is the mean of the dataset,
σ is the standard deviation of the dataset.

a) More than 20 min?

First, we calculate the z-score for 20 minutes:

Z = (20 - 15) / 3.5 = 1.43

We then look up this z-score in the z-table, which gives us the probability that a value is less than 20 minutes. The value for 1.43 in the z-table is 0.9236. However, we want the probability that a listener will tune in for more than 20 minutes, so we subtract the value from 1:

P(X > 20) = 1 - P(X < 20) = 1 - 0.9236 = 0.0764

So, the probability that a listener will tune in for more than 20 minutes is 0.0764, or 7.64%.

b) For 20 min or less?

We already found the probability that a listener will tune in for less than 20 minutes when we looked up the z-score in the z-table. So, the probability that a listener will tune in for 20 minutes or less is 0.9236, or 92.36%.

c) Between 10 and 12 min?

First, we calculate the z-scores for 10 and 12 minutes:

Z(10) = (10 - 15) / 3.5 = -1.43
Z(12) = (12 - 15) / 3.5 = -0.86

We then look up these z-scores in the z-table. The value for -1.43 is 0.0764 and the value for -0.86 is 0.1949. To find the probability that a listener will tune in for between 10 and 12 minutes, we subtract the two probabilities:

P(10 < X < 12) = P(X < 12) - P(X < 10) = 0.1949 - 0.0764 = 0.1185

So, the probability that a listener will tune in for between 10 and 12 minutes is 0.1185, or 11.85%.

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