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. We can say Player A falls close to the average, Player B is above average, and Player C is below average. We can say Player A falls close to the average, Player B is below average, and Player C is above average. We can say Player A and Player B fall close to the average, while Player C is above average. 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. Yes. According to the central limit theorem, when n ≥ 30, the x distribution is approximately normal. Yes. According to the central limit theorem, when n ≤ 30, the x distribution is approximately normal. No. According to the central limit theorem, when n ≥ 30, the x distribution is approximately normal. No. According to the central limit theorem, when n ≤ 30, the x distribution is approximately normal.

Answer 1

Answer:

a) xbar = 2.293
s = 1.401

b)xbar +/- t * s / sqrt(n)

2.293 - 1.684 * 1.401 / sqrt(43) = 1.933

2.293 - 1.684 * 1.401 / sqrt(43) = 2.653

(1.933, 2.653)

Find the t-value that allows us to be 90% confident, go across from

df = 43-1 = 42

(1-.90)/2 = .05

So, the t-critical value is 1.684.

Use the TI-83/84, it will construct the following CI using df = 42 (ie t = 1.681).

=2.293 +/- 1.681 * 1.401 / sqrt(43)

= (1.934, 2.652)

For 90% CI is:

=2.293 +/- 1.645 * 1.401 / sqrt(43)

= (1.942, 2.644)

Z-value that allows us to be 90% confident,

1) Using the z-table, look up (1-.90)/2 = .05 inside the z-table,

2) Using the t-table, go across from infinity df (= z-values) and down from .05 or up from 90% depending on your t-table. Either way,

The z-critical value is 1.645.

c)xbar +/- t * s / sqrt(n)

=2.293 - 2.704 * 1.401 / sqrt(43) = 1.715

=2.293 - 2.704 * 1.401 / sqrt(43) = 2.871

= (1.715, 2.871)

T-value that allows us to be 99% confident, go across from df = 43-1 = 42and down from (1-.99)/2 = .005

So, the t-critical value is 2.704.

Use the TI-83/84, it will construct the following CI using df = 42 (ie t = 2.698).

=2.293 +/- 2.698 * 1.401 / sqrt (43)

= (1.717, 2.869)

Construct a z-CI when the sample size is large. If your professor says this, the correct 99% CI is:

=2.293 +/- 2.576 * 1.401 / sqrt (43)

= (1.742, 2.843)

Z-value that allows us to be 99% confident,

1) Using the z-table, look up (1-.99)/2 = .005 inside the z-table

2) Using the t-table, go across from infinity df (= z-values) and down from .005 or up from 99% depending on your t-table.

Either way, the z-critical value is 2.576.

d)Player A=2.5

Since 2.5 falls between (1.715, 2.871), we see that Player A falls in the 99% CI range. So, his home run percentage is NOT significantly different than the population average.

Player B=2.0

Since 2.0 falls between (1.715, 2.871), we see that Player B falls in the 99% CI range. So, his home run percentage is NOT significantly different than the population average.

Player C=3.8.

Since 3.8 falls above (1.715, 2.871), we see that Player C falls in the 99% CI range. So, his home run percentage IS significantly GREATER than the population average.

e)Because of the Central Limit Theorem (CLT), since our sample size is large, we do NOT have to make the normality assumption since the CLT tells us that the sampling distribution of xbar will be approximately normal even if the underlying population distribution is not.