Question 629065: Suppose that we will randomly select a sample of 64 measurements from a population having a mean equal to 20 and a standard deviation equal to 4.
a. Describe the shape of the sampling distribution of the sample mean x bar. Do we need to make any assumptions about the shape of the population? Why or why not?
b. Find the mean and the standard deviation of the sampling distribution of the sample mean x bar.
c. Calculate the probability that we will obtain a sample mean greater than 21; that is, calculate P(x bar>21). Hint: Find the z value corresponding to 21 by using ?, and ? because we wish to calculate a probability about x bar. Then sketch the sampling distribution and the probability.
d. Calculate the probability that we will obtain a sample mean less than 19.385; that is, calculate P(x bar <19.385).
Answer by stanbon(75887)   (Show Source):
You can put this solution on YOUR website! Suppose that we will randomly select a sample of 64 measurements from a population having a mean equal to 20 and a standard deviation equal to 4.
a. Describe the shape of the sampling distribution of the sample mean x-bar.
Approaching normal.
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Do we need to make any assumptions about the shape of the population? Why or why not?
Ans: Independence of the data.
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b. Find the mean and the standard deviation of the sampling distribution of the sample mean x-bar.
mean of the sample means = mean of the population
std of the sample means = s/sqrt(n)
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c. Calculate the probability that we will obtain a sample mean greater than 21; that is, calculate P(x bar>21). Hint: Find the z value corresponding to 21 by using ?, and ? because we wish to calculate a probability about x-bar. Then sketch the sampling distribution and the probability.
z(21) = (21-20)/(4/sqrt(64)) = 1/(1/2) = 2
P(x-bar < 21) = P(z < 2) = 0.9772
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d. Calculate the probability that we will obtain a sample mean less than 19.385; that is, calculate P(x bar <19.385).
z(19.385) = (19.385-20)/[4/8] = 0.385/0.5 = 0.77
P(x bar <19.385) = P(z < 0.77) = 0.7794
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Cheers,
Stan H.
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