Question

The home run percentage is the number of home runs per 100 times at bat. A random sample of 43 professional baseball players gave the following data for home run percentages.

1.6	2.4	1.2	6.6	2.3	0.0	1.8	2.5	6.5	1.8
2.7	2.0	1.9	1.3	2.7	1.7	1.3	2.1	2.8	1.4
3.8	2.1	3.4	1.3	1.5	2.9	2.6	0.0	4.1	2.9
1.9	2.4	0.0	1.8	3.1	3.8	3.2	1.6	4.2	0.0
1.2	1.8	2.4

(a) Use a calculator with mean and standard deviation keys to find x and s. (Round your answers to two decimal places.)

x =	_______%
s =	_______%

(b) Compute a 90% confidence interval for the population mean μ of home run percentages for all professional baseball players. Hint: If you use the Student's t distribution table, be sure to use the closest d.f. that is smaller. (Round your answers to two decimal places.)

lower limit	______ %
upper limit	______%

(c) Compute a 99% confidence interval for the population mean μ of home run percentages for all professional baseball players. (Round your answers to two decimal places.)

lower limit	______%
upper limit	______%

(d) The home run percentages for three professional players are below.

Player A, 2.5

Player B, 2.0

Player C, 3.8

Examine your confidence intervals and describe how the home run percentages for these players compare to the population average.

A.) We can say Player A falls close to the average, Player B is above average, and Player C is below average.

B.) We can say Player A falls close to the average, Player B is below average, and Player C is above average.    

C.) We can say Player A and Player B fall close to the average, while Player C is above average.

D.) We can say Player A and Player B fall close to the average, while Player C is below average.

(e) In previous problems, we assumed the x distribution was normal or approximately normal. Do we need to make such an assumption in this problem? Why or why not? Hint: Use the central limit theorem.

A.) Yes. According to the central limit theorem, when n ≥ 30, the x distribution is approximately normal.

B.) Yes. According to the central limit theorem, when n ≤ 30, the x distribution is approximately normal.    

C.) No. According to the central limit theorem, when n ≥ 30, the x distribution is approximately normal.

D.) No. According to the central limit theorem, when n ≤ 30, the x distribution is approximately normal.

Answer 1

We can find the value of mean and standard deviation as well as confidence interval using TI-84 calculator.

First enter the all given numbers in the data set into L1 list of TI-84

Press STAT key --> Select 1:Edit and hit enter.

Then press STAT ---> Scroll to CALC ---> Select 1-Var Stats and hit enter.

For List plug L1 ( press 2ND key then 1 ) , Then directly scroll to calculate and hit enter.

a)

= 2.29%

S = 1.40%

b) 99% confidence interval for the population mean μ

Press STAT key ---> Scroll to TESTS --- > Scroll down to T interval and hit enter.

Select Data and hit enter, For List plug L1 ( Press 2ND key then 1)

Then directly scroll to C-Level , plug 0.90

Scroll to calculate and hit enter

lower limit	1.93 %
upper limit	2.65%

c) Compute a 99% confidence interval for the population mean μ

lower limit	1.72 %
upper limit	2.87%

Player A, 2.5

Player B, 2.0

Player C, 3.8

(d) The home run percentages for three professional players are below.

Since player A and B home run perecentages fall within the both interval , but Player C's percentage fall about the interval

Answer : C.) We can say Player A and Player B fall close to the average, while Player C is above average.

(e) In previous problems, we assumed the x distribution was normal or approximately normal. Do we need to make such an assumption in this problem? Why or why not? Hint: Use the central limit theorem.

Answer : A.) Yes. According to the central limit theorem, when n ≥ 30, the x distribution is approximately normal.