Question

The mean amount purchased by each customer at Churchill’s Grocery Store is $28 with a standard deviation of $7. The population is positively skewed. For a sample of 49 customers, answer the following questions:

a. What is the likelihood the sample mean is at least $29? (Round the z-value to 2 decimal places and the final answer to 4 decimal places.)

Sample mean            

b. What is the likelihood the sample mean is greater than $26 but less than $29? (Round the z-value to 2 decimal places and the final answer to 4 decimal places.)

Sample mean            

c. Within what limits will 95% of the sample means occur? (Round the final answers to 2 decimal places.)

Sample mean             and

Answer 1

Solution:

Given:

The mean amount purchased by each customer at Churchill’s Grocery Store is $28 with a standard deviation of $7.

That is: Mean = and standard deviation =

The population is positively skewed.

Sample size = n = 49

Part a) What is the likelihood the sample mean is at least $29?

Sample mean =

We have to find:

Since sample size = n = 49 > 30 , we can assume large sample and hence applying central limit theorem, sampling distribution of sample mean is approximately Normal with mean of sample means is:

and standard deviation of sample means is:

Now find z score for sample mean=

z score formula is:

Thus we get:

Look in z table for z = 1.0 and 0.00 and find area.

From z table , we get:

P( Z < 1.00) = 0.8413

Thus

Part b) What is the likelihood the sample mean is greater than $26 but less than $29?

Sample mean = and

Thus we have to find:

z score:

Thus we get:

We have:

P( Z < 1.00) = 0.8413

Now look for z = -2.0 and 0.00 and find area.

P( Z < -2.00) = 0.0228

Thus we get:

Part c) Within what limits will 95% of the sample means occur?

First find z scores for middle 95% area.

Find Area = ( 1 + c ) / 2 = ( 1 + 0.95) /2 = 1.95 / 2 = 0.9750

Look in z table for Area = 0.9750 or its closest area and find z value.

Area = 0.9750 corresponds to 1.9 and 0.06 , thus z critical value = 1.96

That is : z = 1.96

Since we have middle 95% area , so z value for left tail would be -1.96 and z value for right tail = 1.96

Now use following formula:

and

Thus we get: Sample means to .