Question

The data set below contains 100 records of heights and weights for some current and recent Major League Baseball (MLB) players.
Note: BMI 18.5 - 24.9 normal group, 25 - 29.9 overweight group and > 30 obese group. 

Use the data set to answer the following questions in order:

1.A researcher believes that there is a difference between the BMI of players in the National League vs American League. At a 5% level of significance, is there enough evidence to support the researcher’s claim. (Justify your response by conducting a pooled 2-Sample Mean Hypothesis T-Test.)

2.A researcher believes that a new dietary program can reduce BMI of MLB players. Sixteen players were randomly selected across the MLB league and participated in the new diet program. The 16 MLB players’ BMI was calculated before they started the program and then after 6 months. (see Excel Data File) At a 10% level of significance, is there enough evidence to suggest that the new dietary program for MLB players reduces BMI?

American League
Height(inches) Weight(pounds) Age- BMI
74 180 23 23.1
74 185 23 23.8
74 160 26 20.5
69 180 28 26.6
70 185 34 26.5
73 180 27 23.7
72 188 31 25.5
77 220 33 26.1
74 210 33 27.0
70 195 31 28.0

Diet Program
Height(inches) Weight(pounds) before Weight(pounds) After BMI Before BMI After
73 211 209 27.8 27.6
73 200 193 26.4 25.5
70 180 183 25.8 26.3
70 190 192 27.3 27.5
70 170 166 24.4 23.8
76 230 225 28.0 27.4
68 155 168 23.6 25.5
71 185 190 25.8 26.5
72 185 178 25.1 24.1
75 200 192 25.0 24.0
75 225 222 28.1 27.7
75 225 232 28.1 29.0
75 220 218 27.5 27.2
68 160 176 24.3 26.8
74 205 200 26.3 25.7
78 235 219 27.2 25.3

National League
Height(inches) Weight(pounds) Age BMI
76 230 27 28.0
68 155 26 23.6
71 185 26 25.8
72 185 28 25.1
75 200 25 25.0
75 225 33 28.1
75 225 35 28.1
75 220 31 27.5
68 160 29 24.3
74 205 29 26.3
78 235 28 27.2
71 250 34 34.9
73 210 31 27.7

height Weight(pounds) Age
70 195 25
74 180 23
74 215 35
72 210 31
72 210 35
73 188 36
69 176 29
69 209 31
71 200 35
76 231 30
71 180 27
73 188 24
73 180 27
74 185 23
74 160 26
69 180 28
70 185 34
72 197 30
73 189 28
75 185 22
78 219 23
79 230 26
76 205 36
74 230 31
76 195 32
72 180 31
71 192 29
75 225 29
77 203 32
74 195 36
73 182 26
74 188 27
78 200 24
73 180 27
75 200 25
73 200 28
75 245 30
75 240 31
74 215 31
69 185 32
71 175 28
74 199 28
73 200 29
73 215 24
76 200 22
74 205 25
74 206 27
70 186 33
72 188 31
77 220 33
74 210 33
70 195 31
76 244 37
75 195 26
73 200 23
75 200 25
76 212 24
76 224 35
78 210 27
74 205 31
74 220 28
76 195 30
77 200 25
81 260 24
78 228 30
75 270 26
77 200 23
75 210 26
76 190 25
74 220 32
72 180 24
72 205 25
75 210 24
73 220 24
73 211 32
73 200 30
70 180 24
70 190 32
70 170 23
76 230 27
68 155 26
71 185 26
72 185 28
75 200 25
75 225 33
75 225 35
75 220 31
68 160 29
74 205 29
78 235 28
71 250 34
73 210 31
76 190 38
74 160 24
74 200 26
79 205 24
75 222 24
73 195 28
76 205 33
74 220 36

Answer 1

1)

Null Hypothesis - H₀ : =

Alternate Hypothesis - H₁ :

Use the following data to test the hypothesis -

American BMI	National BMI
23.1	28
23.8	23.6
20.5	25.8
26.6	25.1
26.5	25
23.7	28.1
25.5	28.1
26.1	27.5
27	24.3
28	26.3
	27.2
	34.9
	27.7

Use excel data analysis tool to do the pooled sample t-test to get the following output -

t-Test: Two-Sample Assuming Equal Variances
	American BMI	National BMI
Mean	25.08	27.04615385
Variance	5.132888889	7.914358974
Observations	10	13
Pooled Variance	6.722300366
Hypothesized Mean Difference	0
df	21
t Stat	-1.802877831
P(T<=t) one-tail	0.042888859
t Critical one-tail	1.720742903
P(T<=t) two-tail	0.085777718
t Critical two-tail	2.079613845

As the two tailed p-value is = 0.0858 which is greater than 0.05, so we fail to reject the null hypothesis and conclude that there is not enough evidence to conclude that the overall average BMI differs for American and National players.

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2)

Note that now we have paired data as same 16 people are used to collect the data before and after 6 months.

Use following data for paired sample t-test -

BMI Before	BMI After
27.8	27.6
26.4	25.5
25.8	26.3
27.3	27.5
24.4	23.8
28	27.4
23.6	25.5
25.8	26.5
25.1	24.1
25	24
28.1	27.7
28.1	29
27.5	27.2
24.3	26.8
26.3	25.7
27.2	25.3

Null Hypothesis - H₀ : = 0

Alternate Hypothesis- H₁ : 0

The output from excel is -

t-Test: Paired Two Sample for Means
	BMI Before	BMI After
Mean	26.29375	26.24375
Variance	2.227291667	2.243958333
Observations	16	16
Pearson Correlation	0.713731545
Hypothesized Mean Difference	0
df	15
t Stat	0.176776695
P(T<=t) one-tail	0.431024275
t Critical one-tail	1.753050356
P(T<=t) two-tail	0.86204855
t Critical two-tail	2.131449546

Note that the p-value for one tailed test = 0.43 which is much greater than 0.1, so we fail to reject null hypothesis and conclude that the diet didn't improve lose weight.