Question

1. Murphy’s Law, a pub in downtown Rochester, claims its patrons average 25 years of age. A random sample of 40 bar patrons is taken and their mean age is found to be 26.6 years with a standard deviation of 4.5 years. Do we have enough evidence to conclude at the level of significance α = 0.025 that, on average, patrons at Murphy’s Law are older than 25?

Determine the test statistic.

Determine the range of P-values.

Write your decision and explain how you reached it.

Write the conclusion that addresses the original claim.

Determine the type of error you could have made and explain why. (Type I or Type II

error)

2. The coffee machine at your company has been acting weirdly these past few days. You were assured that the machine would pour 7 ounces of coffee every time you press the button. You believe that this is not true anymore and want to call the technician. Before you do so, you gather a sample of 15 coffees and find out that the mean amount of coffee in each cup is 6.8 ounces with a standard deviation of 1 ounce. Is there sufficient evidence to show that the population mean is different than 7 ounces? Perform a test at the 0.05 level of significance. Assume normality.

Determine the test statistic.

Determine the range of P-values.

Write your decision and explain how you reached it.

Write the conclusion that addresses the original claim.

Determine the type of error you could have made and explain why. (Type I or Type II

error)

Answer 1

Solution:-

1)

State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.

Null hypothesis: u < 25
Alternative hypothesis: u > 25

Note that these hypotheses constitute a one-tailed test. The null hypothesis will be rejected if the sample mean is too small.

Formulate an analysis plan. For this analysis, the significance level is 0.01. The test method is a one-sample t-test.

Analyze sample data. Using sample data, we compute the standard error (SE), degrees of freedom (DF), and the t statistic test statistic (t).

SE = s / sqrt(n)

S.E = 0.7115
DF = n - 1

D.F = 39
t = (x - u) / SE

t = 2.25

where s is the standard deviation of the sample, x is the sample mean, u is the hypothesized population mean, and n is the sample size.

The observed sample mean produced a t statistic test statistic of 2.25.

Thus the P-value in this analysis is 0.015

Interpret results. Since the P-value (0.015) is less than the significance level (0.025), we have to reject the null hypothesis.

From the above test we have sufficient evidence in the favor of the claim that on average, patrons at Murphy’s Law are older than 25.

We could have made Type I error, becuase we are rejecting the null hypothesis.