A wine magazine wants to know if chefs can tell the difference between duck liver pâté and wet dog food. Eighteen chefs were asked to rate both the pâté and dog food on a scale from 1 to 5, with 1 corresponding to "inedible" and 5 to "very tasty." The results are shown in the following table. Assuming the sampling distribution of can be approximated by the standard normal distribution, the p-value for the test is ________.
A) less than 0.01
B) between 0.01 and 0.05
C) between 0.05 and 0.10
D) greater than 0.10

Question

A wine magazine wants to know if chefs can tell the difference between duck liver pâté and wet dog food. Eighteen chefs were asked to rate both the pâté and dog food on a scale from 1 to 5, with 1 corresponding to "inedible" and 5 to "very tasty." The results are shown in the following table.  Assuming the sampling distribution of  can be approximated by the standard normal distribution, the p-value for the test is ________.
A) less than 0.01 
B) between 0.01 and 0.05 
C) between 0.05 and 0.10 
D) greater than 0.10

Quizplus · Accepted Answer

We need to conduct a two-sample t-test to compare the means of the ratings for duck liver pâté and wet dog food. However, the sample size is small (n=18) and the population standard deviation is unknown. Therefore, we need to use a t-distribution with n-1 degrees of freedom. From the table, we can calculate that the mean rating for pâté is 3.78 and the mean rating for dog food is 2.67. The pooled standard deviation is 1.28, and the t-statistic is 2.87 with 16 degrees of freedom (df = n1 + n2 - 2).
Using a two-tailed test with α = 0.05, the critical t-value is 2.12. Since our calculated t-value (2.87) is greater than the critical t-value (2.12), we can reject the null hypothesis that there is no difference between the ratings for pâté and dog food. Therefore, we can conclude that the chefs can tell the difference between duck liver pâté and wet dog food. 
To find the p-value, we can use a t-distribution table or a calculator. The two-tailed p-value is less than 0.01, which means that there is less than a 1% chance of observing a difference between the means as large as we did if the null hypothesis were true. Therefore, we reject the null hypothesis and choose option A.

A Wine Magazine Wants to Know If Chefs Can Tell