According to a survey, high school girls average 100 text messages daily (The Boston Globe, April 21, 2010). Assume the population standard deviation is 20 text messages. Suppose a random sample of 50 high school girls is taken.

a. What is the probability that the sample mean is less than 95?

b. What is the probability that the sample mean is between 95 and 105?

Step-by-step explanation:

Hello!

The variable of interest is:

X: number of daily text messages a high school girl sends.

This variable has a population standard deviation of 20 text messages.

A sample of 50 high school girls is taken.

The is no information about the variable distribution, but since the sample is large enough, n ≥ 30, you can apply the Central Limit Theorem and approximate the distribution of the sample mean to normal:

X[bar]≈N (μ; δ²/n)

This way you can use an approximation of the standard normal to calculate the asked probabilities of the sample mean of daily text messages of high school girls:

Z = (X[bar]-μ) / (δ/√n) ≈ N (0; 1)

a.

P (X[bar]<95) = P (Z< (95-100) / (20/√50)) = P (Z<-1.77) = 0.03836

b.

P (95≤X[bar]≤105) = P (X[bar]≤105) - P (X[bar]≤95)

P (Z≤ (105-100) / (20/√50)) - P (Z≤ (95-100) / (20/√50)) = P (Z≤1.77) - P (Z≤-1.77) = 0.96164-0.03836 = 0.92328

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