Question

# Independent random samples of professional football and basketball players gave the following information. Assume that the...

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 μ1μ2. (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.

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

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.

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