# Final Practice Solutions

## for Stat 185

Here are the solutions to the practice problems for next week's final exam.

## Problem 1

Suppose we randomly select 105 competitors in the 2012 Boston Marathon found an average time of 253.77 minutes with a standard deviation of 45.72 minutes. We wish to write down a $98\%$ confidence interval for this data.

1. Find the standard error associated with this sample.
2. Use a normal table to find the $z^*$ value that corresponds to a $98\%$ confidence interval.
3. Write down the $98\%$ confidence interval.

### Solution

As we know, the confidence interval should have the form $$\bar{x} \pm z^* SE,$$ where $\bar{x}=253.77$ is the computed mean and the standard error is the computed standard deviation divided by the sample size: $$SE = s/\sqrt{n} = 45.72/\sqrt{105} \approx 4.46$$ That last part is the solution to part (a).

### Solution (cont)

To find $z^*$ for a $98\%$ confindence level, look for $(1-0.98)/2 = 0.01$ in the normal table. It looks like, we get a value between $2.32$ and $2.33$. (Either of those is fine, though I prefer to round up.) ### Solution wrap-up

For the final answer, we get

$$\bar{x} \pm z^* SE = 253.77 \pm 2.33 \times 4.46.$$

## Problem 2

Suppose we randomly select 4 runners from the 2012 Boston marathon and find their times in minutes to be

 273.8 203.5 259.4 246.1
1. Write down a formula showing that the mean of these times is $245.7$.
2. Write down a formula showing that the standard deviation of these times is approximately $30.3$.
3. Find the standard error associated with this sample.
4. Write down a $95\%$ confidence interval for the average time of Boston Marathon runner based on this data.

### Solution for part (a)

The mean is $$\bar{x} = \frac{x_1 + x_2 + \cdots + x_n}{n} = \frac{273.8 + 203.5 + 259.4 +246.1}{4} = 245.7.$$

### Solution for part (b)

The standard deviation is \begin{aligned} s &= \sqrt{\frac{(x_1 - \bar{x})^2 + (x_2 - \bar{x})^2 + \cdots + (x_n - \bar{x})^2}{n-1}} \\ &= \sqrt{\frac{(273.8 - 245.7)^2 + (203.5 - 245.7)^2 + (259.5 - 245.7)^2 + (246.1 - 245.7)^2}{3}} \\ &\approx 30.3373. \end{aligned}

### Solution for part (c)

The standard error is $$SE = \frac{s}{\sqrt{n}} = \frac{30.3373}{\sqrt{4}} = 15.16.$$

### Solution to part (d)

To write down the confidence interval, we'll need a $t^*$ multiplier, which we look up in our $t$-table. We find that for a a two-tailed significance of level of $0.05$, we'll need $t^*=3.18$: ### Solution to part (d) wrap-up

Finally, our confidence interval is $$\bar{x} \pm t^* \times SE = 245.7 \pm 3.18 \times 15.15.$$

## Problem 3

A random sample of 1200 runners in the 2012 Boston Marathon found that 507 of them were women. Run a hypothesis test to check the null hypothesis that half of marathon runners are women against the alternative hypothesis that less than half of marathon runners are women. Be sure to

1. Clearly state your hypothesis in terms of differences,
2. compute the standard error,
3. compute the test-statistic, and
4. state the conclusion.

### Solution to part (a)

The hypothesis test is \begin{aligned} & H_0: p = 0.5 \\ & H_A: p < 0.5, \end{aligned} where $p$ denotes the proportion of marathon runners who are women.

### Solution to part (b)

The standard error is $$SE = \sqrt{\frac{p(1-p)}{n}} = \sqrt{\frac{0.5\times0.5}{1200}} \approx 0.0144.$$

### Solution to part (c)

The test-statistic is $$T = \frac{\hat{p} - p}{SE} = \frac{\frac{507}{1200} - 0.5}{0.0144} \approx -5.38$$

To conclude the test, we'll clearly reject the null hypothesis with such an extreme test statistic.

## Problem 4

In the 2012 Boston Marathon, there were

• 7217 runners in their 40s with an average time of 255.2 minutes and a standard deviation of 43.7 minutes, and
• 4156 runners in their 50s with an average time of 270.8 minutes and a standard deviation of 44.7 minutes.

Use this information to test the null hypotesis that $\mu_{40}=\mu_{50}$ against the alternative hypthesis that $\mu_{40} < \mu_{50}$, where $\mu_{40}$ denotes the average time of runners in their forties and $\mu_{50}$ denotes the average time of runners in their fifties. Be sure to

1. Clearly state your hypothesis in terms of differences,
2. compute the standard error,
3. compute the test-statistic, and
4. state the conclusion.

The hypothesis test is \begin{aligned} & H_0: \mu_{40} - \mu_{50} = 0 \\ & H_A: \mu_{40} - \mu_{50} < 0 \\ \end{aligned} where $\mu_{40}$ denotes the average time of marathon runners in their 40s and $\mu_{50} denotes the average time of marathon runners in their 50s. ### Solution for part (b) The standard error is $$SE = \sqrt{\frac{s_{40}^2}{n_{40}} + \frac{s_{50}^2}{n_{50}}} = \sqrt{\frac{43.7^2}{7217} + \frac{44.7^2}{4156}} = 0.863.$$ ### Solution for part (c) The test-statistic is $$\frac{\mu_{40}-\mu_{50}}{SE} = \frac{255.2-270.8}{0.863} = -18.0765.$$ We, again, immediately reject the null hypothesis. ## Problem 5 In the 2012 Boston Marathon, there were 59 runners under the age of 40 who had also run the Boston Marathon in 2002 when they were under the age of 30. I computed the pairwise difference of those runners' times in 2012 minus their times in 2002 and found a mean of 26.1 minutes with a standard deviation of 32.7 minutes. Let's use this data to run a hypothesis test to see if runners slow down over this age range. Specifically, let$\mu_1$denote their first time in 2002 and let$\mu_2$denote their second time in 2012. Test the null hypothesis that$\mu_1 = \mu_2$vs the alternative hypothesis that$\mu_1<\mu_2$at the$99\%confidence level. Be sure to 1. Clearly state your hypothesis in terms of differences, 2. compute the standard error, 3. compute the test-statistic, and 4. state the conclusion. ### Solution to part (a) The hypothesis test is \begin{aligned} & H_0: \mu_{1} - \mu_{2} = 0 \\ & H_A: \mu_{1} - \mu_{2} < 0. \\ \end{aligned} ### Solution for part (b) The standard error is $$SE = s/\sqrt{n} = 32.7/\sqrt{59} = 4.257.$$ ### Solution for part (c) The test statistic is $$\frac{\bar{x}-0}{SE} = \frac{26.1}{4.257} = 6.13.$$ Once again, we immediately reject the null. ## Problem 6 The picture below shows a scatter plot for a random sample of 1200 runners in the Boston Marathon. Thex$-coordinate of each point corresponds to the runner's age and the$y$-coordinate corresponds to the runners time in minutes. The regression line for the data is also shown and has formula$y = 0.719x + 227.6.$ ### Problem 6 (statement continued) 1. What time does this regression model predict for a 56 year old runner? 2. Which of the following could be a reasonable value for the correlation between age and time: 0.9, 0.2, -0.2, or -0.9? 3. Suppose I run a linear regression test on this data and I get results like the following: LinregressResult(slope=0.71868458889427973, intercept=227.5951853339659, rvalue=0.258752071669993, pvalue=8.263097704087485e-20, stderr=0.13124270715070266)  Can I conclude at the 99% level of conficence that there is a linear relationship between age and speed? ### Solution for part (a) Simply plug the number$x=56$into the regression formula to get a prediction of $$0.719\times56 + 227.6 = 267.864.$$ ### Solution for part (b) The positive slope of the regression line indicates a positive relationship between the variables, which rules out the negative numbers. Given how scattered about the points are, I'd have to go with$0.2$over$0.9$. ### Solution for part (c) The outrageously small$p\$-value of

pvalue=8.263097704087485e-20
indicates that we certainly reject the null hypothesis of no relationship, in favor of the alternative hypothesis that there is a relationship between the variables.