The Phillies were accused of cheating their beer customers by not putting enough beer in their 12 oz
cups. When 15 cups were selected at random, they were found to have a mean of 11.7 ounces with a standard deviation of 0.5 ounces. At the .01 significance level, test the claim that the customers are being cheated.
Z = (mean-mean)/SEm
SEm = SD/√n
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability related to Z. Is it ≤ .01?
To test the claim that the customers are being cheated, we can perform a hypothesis test. The null hypothesis (H0) would state that the mean amount of beer in the cups is equal to 12 ounces, while the alternative hypothesis (Ha) would claim that the mean amount of beer in the cups is less than 12 ounces.
Here's how you can perform the hypothesis test to determine if the Phillies are cheating their customers:
Step 1: State the hypotheses:
H0: μ = 12 (Null hypothesis)
Ha: μ < 12 (Alternative hypothesis)
Step 2: Set the significance level:
The significance level, also known as alpha (α), is the probability of rejecting the null hypothesis when it is true. In this case, the significance level is given as 0.01, or 1%.
Step 3: Calculate the test statistic:
To perform the hypothesis test, we will use a t-test since the population standard deviation is unknown. The test statistic for a t-test is calculated using the formula:
t = (x̄ - μ) / (s / √n)
Where:
x̄ = sample mean
μ = hypothesized mean
s = sample standard deviation
n = sample size
In this case, the sample mean (x̄) is 11.7 ounces, the hypothesized mean (μ) is 12 ounces, the sample standard deviation (s) is 0.5 ounces, and the sample size (n) is 15.
t = (11.7 - 12) / (0.5 / √15)
Step 4: Determine the critical value:
The critical value is a threshold that helps us decide whether to reject the null hypothesis. It is determined by the significance level and the degrees of freedom (df). Since the sample size is 15, the degrees of freedom in this case would be 15 - 1 = 14. Using a t-table or a statistical software, you can find the critical value for a one-tailed test with a significance level of 0.01 and 14 degrees of freedom.
Step 5: Compare the test statistic with the critical value:
If the absolute value of the test statistic is greater than the critical value, we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
Step 6: Make a decision:
Based on the comparison of the test statistic and the critical value, we can make a decision on whether to reject or fail to reject the null hypothesis. If we reject the null hypothesis, it would provide evidence to support the claim that the customers are being cheated.
Please note that you'll need to calculate the actual values to make the comparison in Step 5 and make a decision in Step 6.