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 = 21 249 263 255 251 244 276 240 265 257 252 282 256 250 264 270 275 245 275 253 265 271 Weights (in lb) of pro basketball players: x2; n2 = 19 203 200 220 210 193 215 223 216 228 207 225 208 195 191 207 196 183 193 201 (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 Student's t-distribution was used because σ1 and σ2 are known. 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.

Answer 1

(a)

x1

Sum of terms = 21

Number of terms= 249 + 263 + 255 + 251 + 244 + ⋯+ 271 = 5458

Mean=Sum of terms / Number of terms

= 5458 / 21

= 259.9048

s1 =

data	data-mean	(data - mean)2
249	-10.9048	118.91466304
263	3.0952	9.58026304
255	-4.9048	24.05706304
251	-8.9048	79.29546304
244	-15.9048	252.96266304
276	16.0952	259.05546304
240	-19.9048	396.20106304
265	5.0952	25.96106304
257	-2.9048	8.43786304
252	-7.9048	62.48586304
282	22.0952	488.19786304
256	-3.9048	15.24746304
250	-9.9048	98.10506304
264	4.0952	16.77066304
270	10.0952	101.91306304
275	15.0952	227.86506304
245	-14.9048	222.15306304
275	15.0952	227.86506304
253	-6.9048	47.67626304
265	5.0952	25.96106304
271	11.0952	123.10346304

∑(xi−X¯)²=2831.8095

x2 =

Sum of terms = 19

Number of terms= 203 + 200 + 220 + 210 + 193 + ⋯+ 201 = 3914

Mean=Sum of terms / Number of terms

=3914/19= 206

s2 =

data	data-mean	(data - mean)2
203	-3	9
200	-6	36
220	14	196
210	4	16
193	-13	169
215	9	81
223	17	289
216	10	100
228	22	484
207	1	1
225	19	361
208	2	4
195	-11	121
191	-15	225
207	1	1
196	-10	100
183	-23	529
193	-13	169
201	-5	25

∑(xi−X¯)²=2916

(b)

df = 18
tc = 2.878

upper limit = (259.9-206)+ 2.878*sqrt(11.9^2/21 + 12.8^2/19) = 65.2
lower limit = (259.9-206)- 2.878*sqrt(11.9^2/21 + 12.8^2/19) = 42.6

(c)

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.