Question

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

Left:  6 6 8 6 5 7 5 7 5 5

Right:  6 8 7 7 7 6 8 6 8 7

Assume that both populations follow a normal distribution. What can be concluded at the the αα = 0.05 level of significance level of significance?

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

1. The null and alternative hypotheses would be:   
2.   

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

H1:H1:  Select an answer p1 μ1  Select an answer > ≠ = <  Select an answer p2 μ2  (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 reject accept fail to reject  the null hypothesis.
6. Thus, the final conclusion is that ...
   • The results are statistically insignificant at αα = 0.05, 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 = 0.05, so there is sufficient evidence to conclude that the mean innings per game for the eleven left handed starting pitchers that were looked at is less than the mean innings per game for the eleven right handed starting pitchers that were looked at.
   • The results are statistically insignificant at αα = 0.05, so there is insufficient evidence to conclude that the population mean innings per game for left handed starting pitchers is less than the population mean innings per game for right handed starting pitchers.
   • The results are statistically significant at αα = 0.05, so there is sufficient evidence to conclude that the population mean innings per game for left handed starting pitchers is less than the population mean innings per game for right handed starting pitchers.
7. Interpret the p-value in the context of the study.
   • If the sample mean innings per game for the 10 lefties is the same as the sample mean innings per game for the 10 righties and if another another 10 lefties and 10 righties are observed then there would be a 1.49% chance of concluding that the mean innings per game for the 10 lefties is at least 1 innings less than the mean innings per game for the 10 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 10 lefties and 10 righties are observed then there would be a 1.49% chance that the mean number of innings per game for the 10 lefties would be at least 1 innings less than the mean innings per game for the 10 righties.
   • There is a 1.49% chance that the mean innings per game for the 10 lefties is at least 1 innings less than the mean innings per game for the 10 righties.
   • There is a 1.49% chance of a Type I error.
8. Interpret the level of significance in the context of the study.
   • There is a 5% chance that your team will win whether the starting pitcher is a lefty or a righty. What you really need is better pitchers.
   • There is a 5% 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 10 lefties and 10 righties are observed, then there would be a 5% chance that we would end up falsely concluding that the sample mean innings per game for these 10 lefties and 10 righties differ from each other.
   • 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 10 lefties and 10 righties are observed then there would be a 5% chance that we would end up falsely concluding that the population mean innings per game for the lefties is less than the population mean innings per game for the righties

Answer 1

t-test for the difference between two independent population

a)

Ho :   µ1 - µ2 =   0

b)
Ha :   µ1-µ2 <   0

c)

Level of Significance ,    α =    0.05                  
                          
Sample #1   ---->   Left   
mean of sample 1,    x̅1=   6.00                  
standard deviation of sample 1,   s1 =    1.05                  
size of sample 1,    n1=   10                  
                          
Sample #2   ---->   Right
mean of sample 2,    x̅2=   7.00                  
standard deviation of sample 2,   s2 =    0.82                  
size of sample 2,    n2=   10                  
                          
difference in sample means =    x̅1-x̅2 =    6.0000   -   7.0   =   -1.00  
                          
pooled std dev , Sp=   √([(n1 - 1)s1² + (n2 - 1)s2²]/(n1+n2-2)) =    0.9428                  
std error , SE =    Sp*√(1/n1+1/n2) =    0.4216                  
                          
t-statistic = ((x̅1-x̅2)-µd)/SE = (   -1.0000   -   0   ) /    0.42   =   -2.372
                          
Degree of freedom, DF=   n1+n2-2 =    18                  
  
p-value =        0.0145 [ excel function: =T.DIST(t stat,df) ]               
Conclusion:     p-value <α , Reject null hypothesis          

f)

The results are statistically significant at αα = 0.05, so there is sufficient evidence to conclude that the population mean innings per game for left handed starting pitchers is less than the population mean innings per game for right handed starting pitchers.

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