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	255	251	244	276	240	265	257	252	282
256	250	264	270	275	245	275	253	265	271

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

202	200	220	210	193	215	221	216	228	207
225	208	195	191	207	196	181	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 known.The standard normal distribution was used because σ₁ and σ₂ are unknown.    The Student's t-distribution was used because σ₁ and σ₂ are unknown.The Student's t-distribution was used because σ₁ and σ₂ are known.

Answer 1

	Values ( X )	Σ ( Xi- X̅ )2	Values ( Y )	Σ ( Yi- Y̅ )2
	249	116.64	202	13.69
	261	1.44	200	32.49
	255	23.04	220	204.49
	251	77.44	210	18.49
	244	249.64	193	161.29
	276	262.44	215	86.49
	240	392.04	221	234.09
	265	27.04	216	106.09
	257	7.84	228	497.29
	252	60.84	207	1.69
	282	492.84	225	372.49
	256	14.44	208	5.29
	250.0	96.04	195	114.49
	264	17.64	191	216.09
	270	104.04	207	1.69
	275	231.04	196	94.09
	245	219.04	181	610.09
	275	231.04	193	161.29
	253	46.24	201	22.09
	265	27.04
	271	125.44
Total	5456	2823.24	3909	2953.71

Mean X̅ = Σ Xi / n
X̅ = 5456 / 21 = 259.8
Sample Standard deviation S_X = √ ( (Xi - X̅ )² / n - 1 )
S_X = √ ( 2823.24 / 21 -1 ) = 11.9

Mean Y̅ = ΣYi / n
Y̅ = 3909 / 19 = 205.7
Sample Standard deviation S_Y = √ ( (Yi - Y̅ )² / n - 1 )
S_Y = √ ( 2953.71 / 19 -1) = 12.8

Part a)

x1 =	259.8
s1 =	11.9
x2 =	205.7
s2 =	12.8

Part b)

Confidence interval :-

Critical value t(α/2, DF) = t(0.01 /2, 36 ) = 2.719 ( From t table )

DF = 36

Lower Limit =
Lower Limit = 43.4
Upper Limit =
Upper Limit = 64.8
99% Confidence interval is ( 43.4 , 64.8 )

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.