Question

The Aluminum Association reports that the average American uses 56.8 pounds of aluminum in a year. A random sample of 50 households is monitored for one year to determine aluminum usage. If the population standard deviation of annual usage is 12.2 pounds, what is the probability that the sample mean will be each of the following?

Appendix A Statistical Tables

a. More than 58 pounds
b. More than 57 pounds
c. Between 56 and 58 pounds
d. Less than 55 pounds
e. Less than 49 pounds

(Round the values of z to 2 decimal places. Round your answers to 4 decimal places.)

a. enter the probability that the sample mean will be more than 58 pounds

b. enter the probability that the sample mean will be more than 57 pounds

c. enter the probability that the sample mean will be between 56 and 58 pounds

d. enter the probability that the sample mean will be less than 55 pounds

e. enter the probability that the sample mean will be less than 49 pounds

Answer 1

Let 'X' represent pounds of aluminum used in a year.

=56.8  = 12.2

Central limit theorem states that if the sample size is large ( n > 30) then the distribution of the means of similar sample size will approximately follow normal distribution.

n = 50 > 30, so we can use the CLT.   

substituting the values

N( 56.8, 1.7253)

z-score = ( - 56.8)/ 1.7253

So we convert the sample means to z-scores and then using the normal distribution tables we find the probabilities.

a. enter the probability that the sample mean will be more than 58 pounds

P( > 58) = P( Z > 0.70)

=1 - P( Z < 0.70)

=1 - 0.75804

ans: 0.24196

b. enter the probability that the sample mean will be more than 57 pounds

P( > 57) = P( Z > 0.12)

=1 - P( Z < 0.12)

=1 - 0.54776

ans: 0.45224

c. enter the probability that the sample mean will be between 56 and 58 pounds

P(56 < < 58) = P( < 58) - P( < 56)

=P( Z < 0.70) - P(Z < -0.46)

=0.75804 - [1 - P( Z < 0.46)]

=0.75804 - (1 - 0.67724)

ans: 0.43528

d. enter the probability that the sample mean will be less than 55 pounds

P( < 55) = P( Z < -1.04)

=1 - P( Z < 1.04)

=1 - 0.85083

Ans: 0.14917

e. enter the probability that the sample mean will be less than 49 pounds

P( < 49) = P( Z < -4.52)

=1 - P( Z < 4.52)

=1 - 1

Ans: 0