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

245	262	254	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	192	215	223	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 unknown.The standard normal distribution was used because σ₁ and σ₂ are known.    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.

Answer 1

a) For Football players :  

Sample mean using excel function AVERAGE(), x̅1 = 259.6190 = 259.6

Sample standard deviation using excel function STDEV.S, s1 = 12.1222 = 12.1

Sample size, n1 = 21

For Basketball players :  

Sample mean using excel function AVERAGE(), x̅2 = 205.7895 = 205.8

Sample standard deviation using excel function STDEV.S, s2 = 13.0067 = 13.0

Sample size, n2 = 19

b) 99% Confidence interval for the difference:

At α = 0.01 and df = n1+n2-2 = 38, two tailed critical value, t-crit = T.INV.2T(0.01, 38) = 2.712

S²p = ((n1-1)*s1² + (n2-1)*s2² )/(n1+n2-2) = ((21-1)*12.1222² + (19-1)*13.0067²) / (21+19-2) = 157.4766

Lower Bound = (x̅1 - x̅2) - t-crit*√(S²p*(1/n1 +1/n2)) = (259.619 - 205.7895) - 2.712*√(157.4766*(1/21 + 1/19)) = 43.1

Upper Bound = (x̅1 - x̅2) + t-crit*√(S²p*(1/n1 +1/n2)) = (259.619 - 205.7895) + 2.712*√(157.4766*(1/21 + 1/19)) = 64.6

c) The interval consist of numbers that are all positive.

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

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