Home page for accesible maths 2.11 Inference for other estimators 2.11 Inference for other estimators 2.11.2 Hypothesis testing for nearly normal point estimates

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2.11.1 Confidence intervals for nearly normal point estimates

In Section 2.8, we used the point estimate $\bar{x}$ with a standard error $SE_{\bar{x}}$ to create a 95% confidence interval for the population mean:

\displaystyle\bar{x}\ \pm\ 1.96\times SE_{\bar{x}}

(2.12)

We constructed this interval by noting that the sample mean is within 1.96 standard errors of the actual mean about 95% of the time. This same logic generalizes to any unbiased point estimate that is nearly normal. We may also generalize the confidence level by using a place-holder $z^{\star}$ .

General confidence interval for the normal sampling distribution case A confidence interval based on an unbiased and nearly normal point estimate is $\displaystyle\text{point estimate}\ \pm\ z^{\star}SE$ (2.13) where $z^{\star}$ is selected to correspond to the confidence level, and $S E$ represents the standard error. The value $z^{\star}SE$ is called the margin of error.

Generally the standard error for a point estimate is estimated from the data and computed using a formula. For example, the standard error for the sample mean is

\displaystyle SE_{\bar{x}}=\frac{s}{\sqrt{n}}

In this section, we provide the computed standard error for each example and example without detailing where the values came from. In future chapters, you will learn to fill in these and other details for each situation.

Example 2.11.1

In Exercise 2.7.1, we computed a point estimate for the average difference in run times between men and women: $\bar{x}_{women}-\bar{x}_{men}=18.84$ minutes. This point estimate is associated with a nearly normal distribution with standard error $SE=11.14194$ minutes. What is a reasonable 95% confidence interval for the difference in average run times?

Answer. The normal approximation is said to be valid, so we apply Equation (2.13):

\displaystyle\text{point estimate}\ \pm\ z^{\star}SE\quad\rightarrow\quad 18.8% 4\ \pm\ 1.96\times 11.14194\quad\rightarrow\quad(-2.998202,40.6782)

Thus, we are 95% confident that the men were, on average, between 2.998202 minutes slower and 40.6782 minutes faster than women in the 2013 London Marathon. That is, the actual average difference is plausibly between -2.998202 and 40.6782 minutes with 95% confidence.

Example 2.11.2

Does Example 2.11.1 guarantee that if a husband and wife both ran in the race, the husband would run between -2.998202 and 40.6782 minutes faster than the wife?

Answer. Our confidence interval says absolutely nothing about individual observations. It only makes a statement about a plausible range of values for the average difference between all men and women who participated in the run.

Example 2.11.3

The proportion of women in the LonMar13Samp sample is $\hat{p}=0.3573221$ . This sample meets certain conditions that ensure $\hat{p}$ will be nearly normal, and the standard error of the estimate is $SE_{\hat{p}}=0.04792108$ . Create a 90% confidence interval for the proportion of participants in the 2013 London Marathon who are women.

Answer. We use $z^{\star}=1.645$ (see Exercise 2.8.6), and apply the general confidence interval formula:

\displaystyle\hat{p}\ \pm\ z^{\star}SE_{\hat{p}}\quad\to\quad 0.3573221\ \pm\ % 1.645\times 0.04792108\quad\to\quad(0.2784919,0.4361523)

Thus, we are 90% confident that between 28% and 44% of the participants were women.