Question

The mean amount of ice cream in Brand A's containers is 1.75 quarts with a standard deviation of 0.05 quarts. The distribution of the container's volume is approximately normal.

A) Would it be unusual to randomly select a container that held 1.7 quarts or less? Use statistical reasoning to defend your answer.

B) Would it be unusual ti randomly select 16 containers and find the sample mean to be 1.7 quarts or less? Use statistical reasoning to defend your answer.

Answer 1

Given,

X : Amount of ice cream in Brand A's containers;

X follows normal with mean 1.75 quarts a standard deviation of 0.05 quarts

A.

In general , an event is considered  unusual, if the probability of that event is less than 0.05(or less than 5%)

Probability that a randomly selected container that held less than or equal 1.7 quarts = P(X1.7)

Z-score for 1.7 = (1.7 - Mean)/Standard deviation = (1.7 - 1.75) / 0.05 = -0.05/0.05 = -1

From standard normal tables P(Z   -1) = 0.15866

Probability that a randomly selected container that held less than or equal 1.7 quarts = 0.15866

As this probability : 0.15866 > 0.05 ; The event is not considered unusual.

Would it be unusual to randomly select a container that held 1.7 quarts or less

Ans is No, as the Probability that a randomly selected container that held less than or equal 1.7 quarts is 0.15866 which is higher than 0.05.

B.

If X is random variable which follows a normal with mean and standard deviation then

Then Sampling distribution of sample mean follows a normal with mean and standard deviation ;n : sample size

X follows normal with mean 1.75 quarts a standard deviation of 0.05 quarts

sample mean (sample size : 16) follows a normal with mean 1.75 quarts a standard deviation of 0.0125(0.05/)

Probability that a randomly selected a sample of 16 containers and the sample mean to be less than or equal 1.7 quarts

= P(1.7)

P(1.7)

z-score for 17 = Z-score for 1.7 = (1.7 - Mean)/Standard deviation = (1.7 - 1.75) / 0.0125 = -0.05/0.0125 = -4

P(Z -4 ) = 0.00003

P(1.7) = 0.00003

Probability that a randomly selected a sample of 16 containers and the sample mean to be less than or equal 1.7 quarts = 0.00003

As this probability : 0.00003 > 0.05 ; The event is considered unusual.

Would it be unusual ti randomly select 16 containers and find the sample mean to be 1.7 quarts or less

Ans : Yes, As Probability that a randomly selected a sample of 16 containers and the sample mean to be less than or equal 1.7 quarts : 0.00003 > 0.05