Chapter 25, Problem 16E

To determine

To create a $90\%$ confidence interval for the mean difference in winning times for male and female marathon competitors.

Answer to Problem 16E

We are $90\%$ confident that the mean difference in winning times for male and female marathon competitors is between $-17.4409\text{ and }-16.2626$ .

Explanation of Solution

The table of winning times in a marathon race for males and females are given for different years is given in the question. So, let us find the difference between the means of summer and winter as:

Year	Men	Women	Difference
1978	132.2	152.5	-20.3
1979	131.7	147.6	-15.9
1980	129.7	145.7	-16
1981	128.2	145.5	-17.3
1982	129.5	147.2	-17.7
1983	129	147	-18
1984	134.9	149.5	-14.6
1985	131.6	148.6	-17
1986	131.1	148.1	-17
1987	131	150.3	-19.3
1988	128.3	148.1	-19.8
1989	128	145.5	-17.5
1990	132.7	150.8	-18.1
1991	129.5	147.5	-18
1992	129.5	144.7	-15.2
1993	130.1	146.4	-16.3
1994	131.4	147.6	-16.2
1995	131	148.1	-17.1
1996	129.9	148.3	-18.4
1997	128.2	148.7	-20.5
1998	128.8	145.3	-16.5
1999	129.2	145.1	-15.9
2000	130.2	145.8	-15.6
2001	127.7	144.4	-16.7
2002	128.1	145.9	-17.8
2003	130.5	142.5	-12
2004	129.5	143.2	-13.7
2005	129.5	144.7	-15.2
2006	130	145.1	-15.1

Now, it is also given that:

  $n=29$

The sample mean difference is calculated as:

  $\begin{array}{l}\overline{d}=\frac{-20.3-15.9-\mathrm{....}-15.2-15.1}{29}\\ =-16.8517\end{array}$

And the sample standard deviation of the difference is as:

  $\begin{array}{l}{s}_d=\sqrt{\frac{{(-20.3-(-16.8517))}^2+\mathrm{....}+{(-15.1-(-16.8517))}^2}{29-1}}\\ =1.8650\end{array}$

The degree of freedom is as:

  $\begin{array}{l}df=n-1\\ =12-1\\ =11\end{array}$

And therefore the t -value will be as:

  $t=1.7011$

Thus, the confidence interval is then be as:

  $\begin{array}{l}\overline{d}-t_{\alpha /2}\times \frac{s_d}{\sqrt{n}}\\ =-16.8517-1.7011\times \frac{1.8650}{\sqrt{29}}\\ =-17.4409\\ \overline{d}+t_{\alpha /2}\times \frac{s_d}{\sqrt{n}}\\ =-16.8517+1.7011\times \frac{1.8650}{\sqrt{29}}\\ =-16.2626\end{array}$

Thus, we conclude that we are $90\%$ confident that the mean difference in winning times for male and female marathon competitors is between $-17.4409\text{ and }-16.2626$ .