Question

3) American League baseball teams play their games with the designated hitter rule, meaning that pitchers do not bat. The league believes that replacing the pitcher, typically a weak hitter, with another player in the batting order produces more runs. Using a significance level of a = 0.05, determine if the average number of runs is higher for the American League Following are the average number of runs scored by each team in the 2016 season: American League National League Team Runs per Game Team Runs per Game Baltimore Orioles 4.59 Arizona Diamondbacks 4.64 Boston Red Sox 5.42 Atlanta Braves 4.03 Chicago White Sox 4.23 Chicago Cubs 4.99 Cleveland Indians 4.83 Cincinnati Reds 4.42 Detroit Tigers 4.66 Colorado Rockies 5.22 Houston Astros 4.47 Los Angeles Dodgers 4.48 Kansas City Royals 4.17 Miami Marlins 4.07 4.43 Milwaukee Brewers 4.14 Los Angeles Angels Minnesota Twins 4.46 NY Mets 4.14 4.20 Philadelphia Phillies 3.72 NY Yankees Oakland Athletics 4.03 4.5 Pittsburgh Pirates San Diego Padres 4.74 4.23 Seattle Mariners Tampa Bay Rays 4.15 4.41 Texas Rangers 4.72 San Francisco Giants St Louis Cardinals Washington Nationals 4.81 Toronto Blue Jays 4.69 4.71 a) Give the name of the hypothesis test that would be appropriate for this situation. (1 point) b) State the hypotheses in symbols. Make sure to define your variables. (2 points) c) Use your calculator to perform the appropriate hypothesis test and report the test statistic and D-value. Do not make these calculations by hand. Instead, use the correct command in your graphing calculator and write out what you entered. 3 points) d) Write a full conclusion for this test in the context of the problem. (2 points) e) Find a 90% confidence interval to estimate the difference between the mean number of runs per game in American and National League games. Do not make these calculations by hand. Instead, use the correct command in your graphing calculator and write out what you entered. 3 points) f) Interpret the confidence interval in the context of this problem. (2 points) B) Does this confidence interval support your conclusion in part (d)? Explain. 12 points)

Answer 1

Let us denote the American League by subscript 1 and National League by subscript 2.

a). An independent two sample t-test would be appropriate for this situation.  

b). The Hypothesis:

$H_{0}:\mu_{1}=\mu_{2}$

$H_{1}:\mu_{1}>\mu_{2}$ (Average number of runs is higher for American League)

Where $\mu_{1},\mu_{2}$ are the population means of the American and National League.

c). The t-test statistic is 0.6022 and the p-value is 0.2759 for an one tailed test.

d). Since the p-value>0.05, we fail to reject the null hypothesis. Hence, we conclude that there is not enough evidence to claim that the average run scored by American League is more than the National League.

e)The 90% confidence interval is given by (-0.150,0.314).

f). We are 90% confident that the true population difference of the runs between American League and the National League is between -0.150 and 0.314.

g). Since this interval contains the value o, the confidence interval supports the conclusions in part(d)

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t-Test: Two-Sample Assuming Equal Variances
	Amer	National
Mean	4.519333333	4.437333333
Variance	0.12486381	0.153220952
Observations	15	15
Pooled Variance	0.139042381			s1	0.372884
Hypothesized Mean Difference	0			s2	0.365148
df	28			SE	0.136158
t Stat	0.602241787			t	1.701131
P(T<=t) one-tail	0.275929758			Limit	0.231622
t Critical one-tail	1.701130934			diff	0.082
P(T<=t) two-tail	0.551859515			Lower	-0.14962
t Critical two-tail	2.048407142			Upper	0.313622