A random sample is drawn from a normally distributed population with mean mu = 1

Question

A random sample is drawn from a normally distributed population with mean mu = 12 and standard deviation sigma = 1.5. Use Table 1. a. Are the sampling distributions of the sample mean with n = 20 and n = 40 normally distributed? Yes No b. Can you use the standard normal distribution to calculate the probability that the sample mean is less than 12.5 for both sample sizes? Yes No c. Calculate the above probabilities for both sample sizes. (Round intermediate calculations to 4 decimal places, "z" value to 2 decimal places, and final answer to 4 decimal places.)

Explanation / Answer

a.) Answer Yes

  Reason Larger the sample size, the closer the sampling distribution of mean would be to a normal distribution.

b.) Answer Yes

c.) we have to find P(X < 12.5)

So lets find Z-score for both sample size.

For n=20

Z = (12.5 - 12)÷ (1.5/20)

= 1.49

So, P(Z > 1.49) = 1 - Area to left of Z(=1.49)

= 0.0681 answer

For n = 40

Z = (12.5 - 12) ÷ (1.5/40)

= 2.11

So, P(Z > 2.11) = 1 - Area to left of Z(=2.11)

= 0.0174 answer

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