A movie theater is testing out their soda machine to see if it is dispensing the same amount for their “small-sized” sodas

Question

 A quality control engineer is concerned that the machine is under filling the cups. He draws a random sample of 8 cups and records the following amounts:

12.12     11.35     11.51     11.86     11.75     11.45     11.92     12.22

A) Construct a 90% confidence interval for 𝝁

B) The soda machine is supposed to be filling the cups with 12 ounces of soda. Based on your interval, is the machine under filling the cups? Explain your reasoning.

Answer 1

Compute the mean and standard deviation of your sample.

The critical t-value for a 90% confidence it the t-value for sample size - 1 degrees of freedom and 0.05 tail value.

The confidence interval is mean +/- critical t-value * standard deviation/square root(sample size).

If the resultant interval only contains values that are less than 12, then the machine is under filling the cups.

Answer 2

sample mean = 11.7725

sample standard deviation = 0.316758

n=8

standard error = sample standard deviation divided by square root of n = 0.316758/sqrt(8) = 0.111991

critical value of t distribution when the degrees of freedom = n-1 = 7 and the confidence level is 90% (5% in each tail) = +/- 1.89458

margin of error = critical t * standard error = 1.89458 * 0.111991 = 0.212175

lower bound of confidence interval = sample mean - margin of error = 11.7725 - 0.2122 = 11.560

upper bound of confidence interval = sample mean + margin of error = 11.7725 + 0.2122 =11.985

Since the interval does not include the value of 12 ounces, the machine is not meeting the target.