Marathon Training A marathon is a foot race spanning 26.2 miles. It was one of t
ID: 3267549 • Letter: M
Question
Marathon Training
A marathon is a foot race spanning 26.2 miles. It was one of the original events of the modern Olympics, where it was a men's only event. The women's marathon did not become an olympic event until 1984. The current world record for the men's marathon, 2:03:38, was set by Patrick Makau Musyoki on 09/25/2011. The current world record for the women's marathon, 2:15:25, was set by Paula Radcliffe on 04/13/2003.
Training for a marathon typically lasts for at least 6 months. The training is gradual, with increases in distance about every 2 weeks. About 1 to 3 weeks before the race, the distance run is decreased slightly. The tables below show the marathon training (in minutes) for a sample of 30 male runners and 30 female runners.
Training Times (in minutes) of Male Runners
Training Times (in minutes) of Female Runners
1. Use the sample to find a point estimate for the mean training time of the (a) male runners & (b) female runners. Show your work and explain your reasoning.
2. Find the standard deviation of the training times of the (a) male runners & (b) female runners. Show your work and explain your reasoning.
3. Use the sample to construct a 95% confidence interval for the population mean training time of the (a) male runners & (b) female runners. Interpret the results. How do your results differ from those of the following question? Show your work and explain your reasoning.
4. Use the sample to construct a 95% confidence interval for the population mean training time of all runners. How do your results differ from those for the previous question? Show your work and explain your reasoning.
5. Use the sample to construct a 99% confidence interval for the population mean training time of all runners. How do your results differ from those for the previous question? Show your work and explain your reasoning
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 185Explanation / Answer
1. The point estimate for the population mean training time, mu is xbar=sigma x/n, where, x is the tarining time (in minutes) and n is sample size.
a. Male runners, xbar1=sigma x1/n1=(155+158+...+185)/30=167.90
b. Female runners, xbar2=sigmax2/n2=(178+179+...+203)/30=189
2. The formula to compute standard deviation is:
s=sqrt[1/n-1 sigma (x-xbar)^2], where, n is sample size, x is raw score, and xbar is sample mean.
a. Male runners, s1=sqrt[1/30-1 {(155-167.90)^2+(158-167.90)^2+...+(185-167.90)^2}]=8.36
b. Female runners, s2=sqrt[1/30-1 {178-189)^2+(179-189)^2+...+(203-189)^2}]=7.90
3.The population training time is normally distributed, population standard deviation is unknown, therefore, use Student's t distribution to compute 95% confidence interval for 1-population mean, mu.
95% c.i for population mean training time=xbar+-talpha/2(s/sqrt n), where, xbar is sample mean, t is t critical at n-1 df and alpha/2 (alpha=0.05), s is sample standard devaition, n is sample size.
a. 95% c.i for mu1(male)=167.90+-2.364(8.36/sqrt 30)=(164.78,171.02)
b. 95% c.i for mu2(female)=189+-2.364(7.90/sqrt 30)=(186.05,191.95)
4. The 95% c.i for population mean running time for all runners (male and female) is as follows:
xbar+-talpha/2(s/sqrt n) [use previous formulas to ocmpute the xbar and s for 60 samples]
=178.45+-2.300(13.35/sqrt 60) (t critical at alpha/2 and 59 degrees of freedom is 2.300)
=(175,181.90)
The 95% c.i for training times for all runners has lesser width (6.9) compared to 95% c.i for training time for male runners (6.24) but higher width compared to 95% c.i for training time for female runners (5.9).