Marathon iraining marathon is a foot race with a distance of

Marathon iraining marathon is a foot race with a distance of 26.22 miles. It was one of the original events of the odern Olympics, where it was a men\'s-only event. The women\'s marathon did not become an lympic event until 1984. The Olympic record for the men\'s marathon was set during the 2008 mpics by Samuel Kamau Wanjiru of Kenya, with a time of 2 hours, 6 minutes, 32 seconds. Olympic record for the women\'s marathon was set during the 2000 Olympies by Naoko mara Takahashi of Japan, with a time of 2 hours, 23 minutes, 14 seconds. in distance ab of 30 male runners and 30 female runners. Training for a marathon typically lasts at least 6 months. The training is gradual, with increases tance about every 2 weeks. About 1 to 3 weeks before the race, the distance run is decreased slightly. The stem-and-leaf plots below show the marathon training times (in minutes) for a sample Training Times (in minutes) of Male Runners 1558999 16 000012344589 170113566779 ey: Training Times (in minutes) of Female Runners Key: 1718= 178 17899 18000012346679 190001345566 2000123 EXERCISES Use the sample to find a point estimate for the mean training time of the a) male runners. (b) female runners. 5. Use the sample to construct a confidence interval for the population mean training time of all runners. How do your results differ from those in Exercise 3? Explain 2. Find the standard deviation of the training 6. A trainer wants to estimate the population times for the (a) male runners. (b) female runners. mean running times for both male and female runners within 2 minutes. Determine the minimum sample size required to construct a 99% confidence interval for the population mean training time of (a) male runners. Assume the population se the sample to construct a 95% confid training time of the ence interval for the population mean standard deviation is 8.9 minutes. a) male runners. b) female runners. In (b) female runners. Assume the population standard deviation is 8.4 minutes. erpret the results of Exercise 3

Solution

1.

a)

Interpreting the stem and leaf plot, we actually have these times:
155
158
159
159
159
160
160
160
160
161
162
163
164
164
165
168
169
170
171
171
173
175
176
176
177
177
179
180
181
185

As

X = Sum(x) / n  

Summing the items, Sum(x) =    5037

As n =    30

Thus,  

X =    167.9 [ANSWER]

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b)

We actually have the data set

178
179
179
180
180
180
180
181
182
183
184
186
186
187
189
190
190
190
191
193
194
195
195
196
196
200
200
201
202
203

Getting the mean, X,      
      
X = Sum(x) / n      
Summing the items, Sum(x) =    5670  
As n =    30  
Thus,      
X =    189   [ANSWER]

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2.

a)

Setting up tables,      
x   x - X   (x - X)^2
155   -12.9   166.41
158   -9.9   98.01
159   -8.9   79.21
159   -8.9   79.21
159   -8.9   79.21
160   -7.9   62.41
160   -7.9   62.41
160   -7.9   62.41
160   -7.9   62.41
161   -6.9   47.61
162   -5.9   34.81
163   -4.9   24.01
164   -3.9   15.21
164   -3.9   15.21
165   -2.9   8.41
168   0.1   0.01
169   1.1   1.21
170   2.1   4.41
171   3.1   9.61
171   3.1   9.61
173   5.1   26.01
175   7.1   50.41
176   8.1   65.61
176   8.1   65.61
177   9.1   82.81
177   9.1   82.81
179   11.1   123.21
180   12.1   146.41
181   13.1   171.61
185   17.1   292.41

Thus, Sum(x - X)^2 =    2028.7  
      
Thus, as       
      
s^2 = Sum(x - X)^2 / (n - 1)      
      
As n =    30  
      
s^2 =    69.95517241  
      
Thus,      
      
s =    8.363920876   [ANSWER]

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b)

Setting up tables,      
x   x - X   (x - X)^2
178   -11   121
179   -10   100
179   -10   100
180   -9   81
180   -9   81
180   -9   81
180   -9   81
181   -8   64
182   -7   49
183   -6   36
184   -5   25
186   -3   9
186   -3   9
187   -2   4
189   0   0
190   1   1
190   1   1
190   1   1
191   2   4
193   4   16
194   5   25
195   6   36
195   6   36
196   7   49
196   7   49
200   11   121
200   11   121
201   12   144
202   13   169
203   14   196
      
Thus, Sum(x - X)^2 =    1810  
      
Thus, as       
      
s^2 = Sum(x - X)^2 / (n - 1)      
      
As n =    30  
      
s^2 =    62.4137931  
      
Thus,      
      
s =    7.900240066   [ANSWER]

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