Question

Suppose that we will randomly select a sample of 64 measurements from a population having a mean equal to 18 and a standard deviation equal to 5. (a) Describe the shape of the sampling distribution of the sample mean Picture. Do we need to make any assumptions about the shape of the population? Why or why not? ; , because the sample size is . (b) Find the mean and the standard deviation of the sampling distribution of the sample mean Picture. (Round your formula209.mml answer to 1 decimal place.) µPicture 18 ?Picture .625 (c) Calculate the probability that we will obtain a sample mean greater than 20; that is, calculate P(Picture > 20). Hint: Find the z value corresponding to 20 by using µPicture and ?Picture because we wish to calculate a probability about Picture. (Round probability to four decimals) P(formula31.mml > 20) (d) Calculate the probability that we will obtain a sample mean less than 17.564; that is, calculate P(Picture < 17.564) (Round probability to four decimals) P(formula31.mml < 17.564)

Answer 1

Solution:

We are given that : a sample of 64 measurements is selected from a population having a mean equal to 18 and a standard deviation equal to 5. That is : and n = 64

Part a) Describe the shape of the sampling distribution of the sample mean .

the shape of the sampling distribution of the sample mean is approximate Normal distribution with mean of sample means is = and Standard deviation of sample means is =

Do we need to make any assumptions about the shape of the population?

Yes. because the sample size is 64 > 30.

Part (b) Find the mean and the standard deviation of the sampling distribution of the sample mean.

Mean of the sampling distribution of the sample mean is :

the standard deviation of the sampling distribution of the sample mean =

Part c) Calculate the probability that we will obtain a sample mean greater than 20

That is :

find z score :

Thus we get :

Look in z table for z = 3.2 and 0.00 and find area.

P( Z < 3.20 ) = 0.9993

Thus

Part d) Calculate the probability that we will obtain a sample mean less than 17.564

That is :

find z score :

Look in z table for z= -0.7 and 0.00 and find area.

P( Z < -0.70 ) = 0.2420

Thus