Question

A leading consumer magazine, The False Traders, claims that the large coffee sold at PHILZ coffee shop does not contain 16 ounces of coffee. A sample of 16 customers who purchased a large coffee yielded the following information: - a sample mean equal to 15.85 ounces and a sample standard deviation of 0.16 ounces.

a. Set up a 99% confidence interval for the true population mean?

b. At the 1% level, test the hypothesis whether the true mean is less than 16 ounces.

Answer 1

a)
sample mean, xbar = 15.85
sample standard deviation, s = 0.16
sample size, n = 16
degrees of freedom, df = n - 1 = 15

Given CI level is 99%, hence α = 1 - 0.99 = 0.01
α/2 = 0.01/2 = 0.005, tc = t(α/2, df) = 2.947

CI = (xbar - tc * s/sqrt(n) , xbar + tc * s/sqrt(n))
CI = (15.85 - 2.947 * 0.16/sqrt(16) , 15.85 + 2.947 * 0.16/sqrt(16))
CI = (15.73 , 15.97)

b)
Below are the null and alternative Hypothesis,
Null Hypothesis, H0: μ = 16
Alternative Hypothesis, Ha: μ < 16

Test statistic,
t = (xbar - mu)/(s/sqrt(n))
t = (15.85 - 16)/(0.16/sqrt(16))
t = -3.75

P-value Approach
P-value = 0.001

As P-value < 0.01, reject the null hypothesis.

Yes true mean is less than 16 ounces.