A machine that is programmed to package 1.20 pounds of cereal is being tested for its accuracy in a sample of 36 cereal boxes, the sample mean filling weight is calculated as 1.22 pounds. The population standard deviation is known to be 0.06 pounds. Which of the following conclusions is correct?
A) The machine is operating improperly because the target is below the lower limit.
B) The machine is operating properly because the interval contains the target.
C) The machine is operating improperly because the target is above the upper limit.
D) There is not enough information to make a conclusion.

Question

Quizplus · Accepted Answer

We can use a one-sample t-test to test the hypothesis that the population mean filling weight is equal to 1.20 pounds.

Ho: μ = 1.20 
Ha: μ ≠ 1.20

The sample mean, x̄ = 1.22 pounds, and the population standard deviation, σ = 0.06 pounds, are given.

We can calculate the t-value as follows:

t = (x̄ - μ) / (σ / sqrt(n)) 
t = (1.22 - 1.20) / (0.06 / sqrt(36)) 
t = 2

Using a t-table with 35 degrees of freedom (n - 1), we find that the critical values for a two-tailed test at a 5% significance level are ±2.03.

Since our calculated t-value (2) falls within this range, we fail to reject the null hypothesis. This means we do not have enough evidence to conclude that the machine is operating improperly because the interval contains the target (1.20 pounds). Therefore, the correct conclusion is choice B, "The machine is operating properly because the interval contains the target."

A Machine That Is Programmed to Package 1