Question

Do left handed starting pitchers pitch more innings per game on average than right handed starting pitchers? A researcher looked at ten randomly selected left handed starting pitchers' games and nine randomly selected right handed pitchers' games. The table below shows the results.

Left:  7 7 5 6 6 6 6 4 5

Right:  6 4 5 6 5 6 4 7

Assume that both populations follow a normal distribution. What can be concluded at the the αα = 0.10 level of significance level of significance?

For this study, we should use Select an answer t-test for the difference between two dependent population means t-test for the difference between two independent population means z-test for a population proportion z-test for the difference between two population proportions t-test for a population mean

1. The null and alternative hypotheses would be:    

H0: Select an answer μ1 p1   Select an answer < = ≠ > Select an answer μ2 p2 (please enter a decimal)   

H1: Select an answer μ1 p1 Select an answer < > ≠ = Select an answer μ2 p2 (Please enter a decimal)

2. The test statistic? t z  =  (please show your answer to 3 decimal places.)
3. The p-value =  (Please show your answer to 4 decimal places.)
4. The p-value is? ≤ >  αα
5. Based on this, we should Select an answer accept fail to reject reject  the null hypothesis.
6. Thus, the final conclusion is that ...
   • The results are statistically insignificant at αα = 0.10, so there is insufficient evidence to conclude that the population mean innings per game for left handed starting pitchers is more than the population mean innings per game for right handed starting pitchers.
   • The results are statistically insignificant at αα = 0.10, so there is statistically significant evidence to conclude that the population mean innings per game for left handed starting pitchers is equal to the population mean innings per game for right handed starting pitchers.
   • The results are statistically significant at αα = 0.10, so there is sufficient evidence to conclude that the mean innings per game for the ten left handed starting pitchers that were looked at is more than the mean innings per game for the nine right handed starting pitchers that were looked at.
   • The results are statistically significant at αα = 0.10, so there is sufficient evidence to conclude that the population mean innings per game for left handed starting pitchers is more than the population mean innings per game for right handed starting pitchers.
7. Interpret the p-value in the context of the study.
   • There is a 21.48% chance that the mean innings per game for the 9 lefties is at least 0.4 innings more than the mean innings per game for the 8 righties.
   • There is a 21.48% chance of a Type I error.
   • If the sample mean innings per game for the 9 lefties is the same as the sample mean innings per game for the 8 righties and if another another 9 lefties and 8 righties are observed then there would be a 21.48% chance of concluding that the mean innings per game for the 9 lefties is at least 0.4 innings more than the mean innings per game for the 8 righties
   • If the population mean innings per game for left handed starting pitchers is the same as the population mean innings per game for right handed starting pitchers and if another 9 lefties and 8 righties are observed then there would be a 21.48% chance that the mean number of innings per game for the 9 lefties would be at least 0.4 innings more than the mean innings per game for the 8 righties.
8. Interpret the level of significance in the context of the study.
   • There is a 10% chance that your team will win whether the starting pitcher is a lefty or a righty. What you really need is better pitchers.
   • If the population mean innings per game for left handed starting pitchers is the same as the population mean innings per game for right handed starting pitchers and if another 9 lefties and 8 righties are observed then there would be a 10% chance that we would end up falsely concluding that the population mean innings per game for the lefties is more than the population mean innings per game for the righties
   • There is a 10% chance that there is a difference in the population mean innings per game for lefties and righties.
   • If the population mean innings per game for lefties is the same as the population mean innings per game for righties and if another 9 lefties and 8 righties are observed, then there would be a 10% chance that we would end up falsely concluding that the sample mean innings per game for these 9 lefties and 8 righties differ from each other.

Answer 1

R OUTPUT:

> left=c(7,7,5,6,6,6,6,4,5)
> right=c(6,4,5,6,5,6,4,7)
> var.test(left,right,alternative="two.sided")

F test to compare two variances

data: left and right
F = 0.83951, num df = 8, denom df = 7, p-value = 0.8041
alternative hypothesis: true ratio of variances is not equal to 1
95 percent confidence interval:
0.1713508 3.8017559
sample estimates:
ratio of variances
0.8395062

> t.test(left,right,mu=0,alternative="greater",var.equal=TRUE,conf.level=0.90)

Two Sample t-test

data: left and right
t = 0.81726, df = 15, p-value = 0.2148
alternative hypothesis: true difference in means is greater than 0
90 percent confidence interval:
-0.2579224 Inf
sample estimates:
mean of x mean of y
5.777778 5.375000

First we have performed variance equality test. F-test shows p-value=0.8041 which implies variances are equal. So, for this study, we should use t-test for the difference between two independent population means.

a)The null and alternative hypotheses would be:    

H0: μ1=μ2 vs  H1: μ1≠μ2

b)t-statistic=0.817

c)p-value=0.2133>0.10

d)Based on this, we should fail to reject the null hypothesis.

e)Final conclusion:

The results are statistically insignificant at αα = 0.10, so there is insufficient evidence to conclude that the population mean innings per game for left handed starting pitchers is more than the population mean innings per game for right handed starting pitchers.

g)Interpretation of p-value:

If the population mean innings per game for left handed starting pitchers is the same as the population mean innings per game for right handed starting pitchers and if another 9 lefties and 8 righties are observed then there would be a 21.48% chance that the mean number of innings per game for the 9 lefties would be at least 0.4 innings more than the mean innings per game for the 8 righties.

h)interpretation of level of significance:

If the population mean innings per game for left handed starting pitchers is the same as the population mean innings per game for right handed starting pitchers and if another 9 lefties and 8 righties are observed then there would be a 10% chance that we would end up falsely concluding that the population mean innings per game for the lefties is more than the population mean innings per game for the righties