Question

The owner of a local golf course wants to estimate the difference between the scores of golfers on his golf course and the scores of golfers on his friend's golf course. He takes a random sample of 24 golfers and has them play a round of golf on his course and he records their scores. The next day, he has the same 24 golfers play a round of golf on his friends course and records the scores again. He uses the scores to calculate a paired t-interval of (-3.31, 14.74) with a confidence level of 90%. Which of the following statements is the appropriate conclusion? The differences were calculated as (score on the owner's course - score on the friend's course).

Question 8 options:

1) We are 90% confident that the average difference in scores for all golfers is negative, with the higher scores occuring on the owner's course.

2) We are 90% confident that the average difference in scores for all golfers is positive, with the higher scores occuring on the owner's course.

3) No significant difference exists between the average difference in scores and 0.

4) We are 90% confident that the average difference in scores for all golfers is positive, with the higher scores occuring on the friend's course.

5) We are 90% confident that the average difference in scores for all golfers is negative, with the higher scores occuring on the friend's course.

2. The owner of a local golf course wants to estimate the difference between the average ages of males and females that play on the golf course. He randomly samples 22 men and 20 women that play on his course. He finds the average age of the men to be 40.108 with a standard deviation of 9.953. The average age of the women was 33.245 with a standard deviation of 6.15. He uses this information to calculate a 99% confidence interval for the differnce in means, (-0.124, 13.85). The best interpretation of this interval is which of the following statements?

Question 2 options:

1) We are 99% confident that the difference between the average age of all men and all women who play golf at the course is between -0.124 and 13.85
2) We are 99% confident that the difference between the average age of the men and women surveyed is between -0.124 and 13.85
3) We are certain that the differece between the average age of all men and all women is between -0.124 and 13.85.
4) We do not know the population means so we do not have enough information to make an interpretation.
5) We are 99% sure that the average age difference between all males and females is between -0.124 and 13.85.

Answer 1

(8) The correct option is (3).

(2) The correct option is (1).

Explanation:

(8)

The hypothesis can be defined as:

H₀: There is no difference between the two population means, i.e. μ₁ = μ₂.

Hₐ: There is a difference between the two population means, i.e. μ₁ ≠ μ₂.

The decision rule of the test based on confidence interval is:

If the (1 - α) % confidence interval consists the null hypothesis value then the null hypothesis is not rejected. and vice-versa.

The 90% confidence interval for the paired difference between means is, (-3.31, 14.74).

The interval consists the null value, 0.

Thus, the null hypothesis was failed to be rejected.

Conclusion:

No significant difference exists between the average difference in scores and the differences were calculated as 0.

Thus, the correct option is (3).

(2)

A (1 - α) % confidence interval implies that there is (1 - α) % chance of the population parameter to be contained in this interval.

Or if 100 such intervals are constructed then (1 - α) of these intervals will consist the true parameter value.

The 99% confidence interval for the difference between means is, (-0.124, 13.85).

This interval implies that the true mean difference between the two populations lies in this interval with probability 0.99.

Conclusion:

We are 99% confident that the difference between the average age of all men and all women who play golf at the course is between -0.124 and 13.85.

Thus, the correct option is (1).