Given the following information for a normally distributed population; Populatio

Question

Given the following information for a normally distributed population;      Population mean (µ = 4.5) and the standard deviation ( = 0.6). Find the following values:

Find the probability of selecting a sample of one (n = 1) from the population with a sample mean greater than 4.7?

Find the probability of selecting a sample of n = 36 with a sample mean greater than 4.7?

Find the probability of selecting a sample of n = 81 with a sample mean greater than 4.7?

Examine the results and describe what happens to the probability as the sample size increases. Why?

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Given that the sample mean (X = 75), the population standard deviation ( = 12), and the desired level of confidence or confidence coefficient is 95%. Establish interval estimates for the following sample sizes:

What is the best point estimate for the population mean?

Sample size n = 36.

Sample size n = 64.

Sample size n = 100.

What happens to the interval estimates as the sample size increases? Why?

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Given that the sample mean (X = 80), the population standard deviation ( = 20), and the sample size is (n = 100). Establish interval estimates for the following the desired level of confidence or confidence coefficients:

What is the best point estimate for the population mean?

Level of confidence is 90%.

Level of confidence is 95%.

Level of confidence is 99%

What happens to the interval estimates as the Level of confidence increases? Why?

Explanation / Answer

One question at a time. Attempting 1st question:

Population mean (µ = 4.5) and the standard deviation ( = 0.6)

1. n=1
P(Xbar>4.7) = P(Z> (4.7-4.5)/(.6/sqrt(1)) = P(Z>.33) = .6293

2.Find the probability of selecting a sample of n = 36 with a sample mean greater than 4.7?
P(Xbar>4.7) = P(Z> (4.7-4.5)/(.6/sqrt(36)) = P(Z>1.98) = 1-.9761 = .0324

3.Find the probability of selecting a sample of n = 81 with a sample mean greater than 4.7?
P(Xbar>4.7) = P(Z> (4.7-4.5)/(.6/sqrt(81)) = P(Z>2.97) = 1-.9985 = .0015

As the sample size increases the distribution turns more "Normal" like. Thus the area under the curve beyond 4.7 decreases.

Examine the results and describe what happens to the probability as the sample size increases. Why?

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