Question

(10 points) The ESPN Home Run Tracker computes a variety of measurements for every home run hit in Major League Baseball, including true distance, speed of bat, elevation angle, and apex. Independent random samples of home runs during the 2013 season were obtained from four different ballparks, and the distance was recorded for each. The summary statistics are given in the following table: Ballpark Fenway Park Coors Field Dodger Stadium Rogers Centre 34 34 34 34 n 397.8 396.2 413.28 400.1 х 21.6 15.88 20.85 26.33 S

1. Assuming that the populations are normal, is there any evidence to suggest that the average home-run distances in at least two of the ballparks are different? Use α = 0.05. Perform the complete four-step summary. Be sure to list all of the values that you have previously calculated for the required four-steps including the two degrees of freedom.

2. Construct the Tukey 95% confidence intervals to isolate the pairs of means contributing to the overall difference. For each pair indicate whether the two means are different or not. Hint: Work on submitting three trials for the Tukey parameter and the standard error first. If you do not get them in three trials, continue submitting up to three trials for each of the confidence intervals until you get one correct. Then go back and submit the fourth trial for the previous parts. Do not submit the one trial for the significance until the interval is correct.

a)Calculate the Tukey parameter. Note that this is not the same as the critical value.

b)Calculate the standard error. Please use three decimal places.

Answer 1

1)

	1	2	3	4
count, ni =	34	34	34	34
mean , x̅ i =	396.200	413.28	397.80	400.10
std. dev., si =	15.880	21.600	20.850	26.330
sample variances, si^2 =	252.174	466.560	434.723	693.269
total sum	13470.8	14051.52	13525.2	13603.4	54650.92	(grand sum)
grand mean , x̅̅ =	Σni*x̅i/Σni =				401.85
square of deviation of sample mean from grand mean,( x̅ - x̅̅)²	31.866	130.759	16.362	3.045
					TOTAL
SS(between)= SSB = Σn( x̅ - x̅̅)² =	1083.445	4445.814	556.309	103.531	6189.098
SS(within ) = SSW = Σ(n-1)s² =	8321.755	15396.480	14345.843	22877.874	60941.951

no. of treatment , k =   4
df between = k-1 =    3
N = Σn =   136
df within = N-k =   132
  
mean square between groups , MSB = SSB/k-1 =    2063.0327
  
mean square within groups , MSW = SSW/N-k =    461.6815
  
F-stat = MSB/MSW =    4.4685

	SS	df	MS	F	p-value	F-critical
Between:	6189.098	3	2063.033	4.47	0.0051	2.67
Within:	60941.951	132	461.681
Total:	67131.050	135
					α =	0.05

Ho: µ1=µ2=µ3=µ4
H1: at least two of the ballparks are different

df1 = 3

df2=132

F stat = 4.47

p-value<α , reject null hypothesis
conclusion: there is enough evidence that at least two of the ballparks are different at α=0.05