Question

Confidence intervals: A random sample of 6 bowlers at a state bowling tournament produced the following scores:

Steve: 185,

Hans: 165,

Brianna: 85,

Samantha: 135,

Paul: 80,

Jennifer: 225.

Assume these 6 bowlers are a representative sample of all bowlers in the tournament.

To answer this question, you will need to calculate:

a. The point estimate for the average score of all bowlers in the tournament, and

b. The point estimate for the standard deviation of scores for all bowlers in the tournament.

(Write down those numbers because you will be asked to data-enter them in the question that follows this question.)

Coach Swanson believes that the scores of all bowlers at the tournament are normally distributed.

What is the 90% confidence interval for the average score of a bowler at this tournament. Use Table 2.

(Round intermediate calculations to 4 decimal places, "sample mean" and "sample standard deviation" to 2 decimal places. Round final answers to 2 decimal places. Provide the low number first, to the high number second.)

Item23

Item 23

In the previous question, you calculated:

a. The point estimate for the average score of all of the bowlers in the tournament, and

b. The point estimate for the standard deviation of all scores for bowlers in the tournament.

What did you calculate for the (a) average score, and for the (b) standard deviation? (Provide those statistics now.)

Answer 1

Solution:

Given:

A random sample of 6 bowlers at a state bowling tournament produced the following scores:

Steve: 185,

Hans: 165,

Brianna: 85,

Samantha: 135,

Paul: 80,

Jennifer: 225.

Part a. The point estimate for the average score of all bowlers in the tournament

Bowlers	x
Steve	185
Hans	165
Brianna	85
Samantha	135
Paul	80
Jennifer	225

Sample mean is the point estimate for the average.

Thus Sample mean is given by:

Part b. The point estimate for the standard deviation of scores for all bowlers in the tournament.

Sample standard deviation is the point estimate for the standard deviation.

Thus we need to make following table:

Bowlers	x	x^2
Steve	185	34225
Hans	165	27225
Brianna	85	7225
Samantha	135	18225
Paul	80	6400
Jennifer	225	50625

Thus

Part c. Coach Swanson believes that the scores of all bowlers at the tournament are normally distributed.

What is the 90% confidence interval for the average score of a bowler at this tournament.

Since sample size n = 6 is small and population standard deviation is unknown we use t distribution to find 90% confidence interval.

where

We need to find t_c value for c = 90% = 0.90 confidence level.

Thus two tailed area = 1 -c = 1 - 0.90 = 0.10

df = n - 1 = 6 - 1 = 5

Look in t table for df = 5 and two tailed area =0.10

and find t critical value.

t critical value = 2.015

that is: t_c = 2.015

Thus margin of error E is:

Thus limits of confidence interval are:

The 90% confidence interval for the average score of a bowler at this tournament is between:

a. The point estimate for the average score of all bowlers in the tournament is

b. The point estimate for the standard deviation of scores for all bowlers in the tournament

c.The 90% confidence interval for the average score of a bowler at this tournament is between: