Question

The Aluminum Association reports that the average American uses 56.8 pounds of aluminum in a year. A random sample of 49 households is monitored for one year to determine aluminum usage. If the population standard deviation of annual usage is 12.3 pounds, what is the probability that the sample mean will be each of the following?

Appendix A Statistical Tables

a. More than 60 pounds
b. More than 56 pounds
c. Between 55 and 57 pounds
d. Less than 55 pounds
e. Less than 48 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 60 pounds

b. enter the probability that the sample mean will be more than 56 pounds

c. enter the probability that the sample mean will be between 55 and 57 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 48 pounds

Answer 1

The average American uses = 56.8 pounds of aluminum in a year. A random sample of n= 49 households is monitored for one year to determine aluminum usage. If the population standard deviation of annual usage is =12.3 pounds.

For a normal distribution the Z score is applicable for probability calculation by finding the Z scores respective to the sample mean values in following questions.

a) For sample mean more than >60 pounds.

The Z score is calculated as:

Thus P(>60)=P(Z>1.82)

The probability is calculated using excel formula for normal distribution, the formula used is =1-NORM.S.DIST(1.82,TRUE) which gives P(>60)=P(Z>1.82)=0.0344

b)For sample mean more than 56 pounds.

The Z score is calculated as:

now,Thus P(>56)=P(Z>-0.46) is calculated again using excel formula for normal distribution, the formula used is =1-NORM.S.DIST(-0.46,TRUE) which results in:

P(>56)=P(Z>-0.46)=0.6772

c) For sample mean between 55 and 57 pounds.

The Z score for sample means 55 and 57 pounds. is calculated as:

So, the probability is calculated as:

Pr(-1.02<Z<0.11) is calculated using excel formula for normal distribution, the formula used for probability calculation by excel is =NORM.S.DIST(0.11,TRUE)-NORM.S.DIST(-1.02,TRUE) which results in probability as:

=>0.3899

d) For sample mean Less than 55 pounds.

The Z score is calculated as:

Now probability is calculated as:

P(<55)=P(Z<-1.02) is again calculated using excel formula for normal distribution by formula =NORM.S.DIST(-1.02,TRUE) which results in

P(<55)=P(Z<-1.02)=0.1539

e) The probability that the sample mean less than 48 pounds.

The Z score is calculated as:

Now the probability is calculated using excel formula =NORM.S.DIST(-5.01,TRUE) which results in

P(<48) is 0.

Note: The probability may vary since i have used excel for accurate calculation