Question

Christiano Ronaldo has informed Nike that he is very particular about putting his name on a new line of soccer cleats. He will not endorse the product unless there is evidence that no more than 2% of the shoes coming off the assembly line have defects. Nike reports that a random sample of 250 shoes contains 6 defective shoes. a. Formulate and test an appropriate set of hypotheses to determine whether Ronaldo should endorse the shoes. Use ?=0.05. Be sure to explicitly state: The null and alternative hypotheses The test statistic equation and value Your conclusion b. What is the smallest level of significance at which you would be willing to reject the null hypothesis (the P-value)? (You may provide a precise value if you are using a calculating device, or a range if you are using the tables.) c. Ronaldo appears confused when you tell him the P-value and says that he is more comfortable with confidence intervals. Formulate a two-sided confidence interval and explain how it supports your conclusion.

Answer 1

Solution:

a. Null Hypothesis (Ho): p 0.02

Alternative Hypothesis (Ha): p > 0.02

Sample proportion, p' = x/n

Sample proportion, p' = 6/250

Sample proportion, p' = 0.024

Test Statistics

Z = (0.024 - 0.02)/0.02*(1 - 0.02)/250

Z = 0.14

Using Z-tables, the critical value at a = 0.05 is 1.645

Since test statistics is less than the critical value, we fail to reject Ho.

Hence, we can conclude that he will endorse the product.

b. Using excel, the exact p-value is 0.4443

Since p-value is greater than 0.05 significance level, we fail to reject Ho.

c. 95% confidence interval is given by:-

p Z (a/2)*p (1 - p)/n

0.024 Z (0.05/2)*0.024 (1 - 0.024)/250

0.024 1.96*0.0000937

0.024 1.96*0.0097

0.024 0.019

0.005, 0.043

Since 0.02 lies within the interval 0.005 to 0.043, we fail to reject Ho.

Hence, we can conclude that he will endorse the product.