On the SAT exam a total of 25 minutes is allotted for students to answer 20 math questions without the use of a calculator. A guidance counselor would like to know if the students in his school are prepared to complete this portion of the exam in the timeframe allotted. To investigate, the counselor selects a random sample of 35 students and administers this portion of the test. The students are instructed to turn in their test as soon as they have completed the questions. The mean amount of time taken by the students is 23 point 5 minutes with a standard deviation of 4 point 8 minutes. The counselor would like to know if the data provide convincing evidence that the true mean amount of time needed for all students of this school to complete this portion of the test is less than 25 minutes and therefore tests the hypotheses ratio of cap h sub 0 to mu is equal to 25 versus ratio of cap h sub A to mu is less than 25 , where mu is equal to the true mean amount of time needed by students at this school to complete this portion of the exam. The conditions for inference are met. What are the appropriate test statistic and cap p -value? Use the t -table.
Answer
A
t is equal to negative 1 point 8 5 ; the cap p -value is between 0 point 0 2 5 and 0 point 0 5 .
B
t is equal to 1 point 8 5 ; the cap p -value is 0 point 9 6 7 8 .
C
t is equal to negative 1 point 8 5 ; the cap p -value is 0 point 9 6 7 8 .
D
t is equal to 1 point 8 5 ; the cap p -value is between 0 point 0 2 5 and 0 point 0 5 .
Explanation
Use a one-sample t statistic because the population standard deviation is not known: t = (23.5 − 25)/(4.8/√35) ≈ −1.85, with 34 degrees of freedom. Since the alternative hypothesis is that the mean is less than 25, the P-value is the area to the left of −1.85. The t-table places this P-value between 0.025 and 0.05, matching the choice with t = −1.85 and that P-value range.
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