Question

Independent random samples of professional football and basketball players gave the following information. Assume that the weight distributions are mound-shaped and symmetric.

Weights (in lb) of pro football players: x₁; n₁ = 21

249	261	254	251	244	276	240	265	257	252	282
256	250	264	270	275	245	275	253	265	270

Weights (in lb) of pro basketball players: x₂; n₂ = 19

203	200	220	210	192	215	222	216	228	207
225	208	195	191	207	196	182	193	201

(a) Use a calculator with mean and standard deviation keys to calculate x₁, s₁, x₂, and s₂. (Round your answers to one decimal place.)

x1 =
s1 =
x2 =
s2 =

(b) Let μ₁ be the population mean for x₁ and let μ₂ be the population mean for x₂. Find a 99% confidence interval for μ₁ − μ₂. (Round your answers to one decimal place.)

lower limit
upper limit

(c) Examine the confidence interval and explain what it means in the context of this problem. Does the interval consist of numbers that are all positive? all negative? of different signs? At the 99% level of confidence, do professional football players tend to have a higher population mean weight than professional basketball players?

Because the interval contains only negative numbers, we can say that professional football players have a lower mean weight than professional basketball players.

Because the interval contains both positive and negative numbers, we cannot say that professional football players have a higher mean weight than professional basketball players.   

Because the interval contains only positive numbers, we can say that professional football players have a higher mean weight than professional basketball players.

(d) Which distribution did you use? Why?

The standard normal distribution was used because σ₁ and σ₂ are unknown.

The Student's t-distribution was used because σ₁ and σ₂ are known.    

The Student's t-distribution was used because σ₁ and σ₂ are unknown.

The standard normal distribution was used because σ₁ and σ₂ are known.

Answer 1

	Football ( X )	Σ ( Xi- X̅ )2	Basketball ( Y )	Σ ( Yi- Y̅ )2
	249	114.49	203	7.84
	261	1.69	200	33.64
	254	32.49	220	201.64
	251	75.69	210	17.64
	244	246.49	192	190.44
	276	265.69	215	84.64
	240	388.09	222	262.44
	265	28.09	216	104.04
	257	7.29	228	492.84
	252	59.29	207	1.44
	282	497.29	225	368.64
	256	13.69	208	4.84
	250	94.09	195	116.64
	264	18.49	191	219.04
	270	106.09	207	1.44
	275	234.09	196	96.04
	245	216.09	182	566.44
	275	234.09	193	163.84
	253	44.89	201	23.04
	265	28.09
	270	106.09
Total	5454	2812.29	3911	2956.56

Mean X̅ = Σ Xi / n
X̅ = 5454 / 21 = 259.7
Sample Standard deviation S_X = √ ( (Xi - X̅ )² / n - 1 )
S_X = √ ( 2812.29 / 21 -1 ) = 11.9

Mean Y̅ = ΣYi / n
Y̅ = 3911 / 19 = 205.8
Sample Standard deviation S_Y = √ ( (Yi - Y̅ )² / n - 1 )
S_Y = √ ( 2956.56 / 19 -1) = 12.8

x₁ =259.7 s₁ = 11.9   x₂ =205.8   s₂ =12.8

Part b)

Confidence interval :-

Critical value   t(α/2, DF) = t(0.01 /2, 36 ) = 2.7195 ( From t table )

DF = 36

Lower Limit =
Lower Limit = 43.2
Upper Limit =
Upper Limit = 64.5
99% Confidence interval is ( 43.2 , 64.5 )

Part c)

Because the interval contains only positive numbers, we can say that professional football players have a higher mean weight than professional basketball players.

Part d)

The Student's t-distribution was used because σ₁ and σ₂ are unknown.