OPTIMA Statistics

Subsection 4.2.1 Point Estimate

Since the goal of this section is to understand how to estimate a population mean, it makes sense to start by asking what the best point estimate is for a mean. That is, if we draw a random sample from a population, and we want to compute one number from that sample that best approximates the mean of the population, what number should we use?

The results of this example are summarized below.

While this may seem obvious, there is some background knowledge that is important. In Section 3.5 we saw that as long as the population was normal, or the sample size was at least 30, then the distribution of sample means, \(\overline{X}\text{,}\) will be approximately normal and have as its mean \(\mu_{\overline{x}}\) the population mean \(\mu\text{.}\) At this point, we don't need to know that the sample means have a normal distribution, although that will be important shortly. What is more important is that

In other words, if we take the average of the sample means \(\overline{x}\) for all possible samples of a set size, that will equal the population mean. So \(\overline{x}\) is an unbiased estimator for \(\mu\text{.}\)

Note that this does not by any means guarantee that \(\overline{x} = \mu\text{.}\) Our sample mean will probably not equal the actual population mean. But it should be consistently close. And if we were to take many different samples, the average of all of their means should be extremely close to the population mean which we are trying to estimate.

Subsection 4.2.2 Margin of Error

Now that we know the best point estimator for a population mean is the sample mean, we have the first piece of our confidence interval. But we need to know how much error we should expect when using this approximation. That is, we need to know how to compute the margin of error. Remember that in general,

SE, where the standard error is the standard deviation in the sampling distribution. From Section 3.5, we know that if the distribution of \(\overline{x}\) is normal, then its standard deviation is \(\frac{\sigma}{\sqrt{n}}\text{.}\) This yields the following.

There are several things to note about this margin of error computation.

• The distribution of the sample means must be normal. That means either the original population must have a normal distribution or we must have a sample size of at least 30 so that we can apply the central limit theorem.

• We must know the value of \(\sigma\text{,}\) the population standard deviation. If we are trying to estimate the population mean, it seems unreasonable that we would already know \(\sigma\text{.}\) However, as long as our sample size is at least 30, we can use the sample standard deviation s as an approximation for \(\sigma\text{.}\)

With these notes in mind, let's look at several examples.

Subsection 4.2.3 Confidence Interval

By simply adding and subtracting the margin of error to the point estimate, we can construct our confidence interval. For a mean, this is done as follows.

Consider the following examples.

In the above example, note what happened when we increased our confidence level. If we only want to be 95% confident that our confidence interval will contain the true mean weight of a burrito, we get a smaller interval than if we want to be 99% confident. Whenever we increase the confidence level, this will also increase the width of our confidence intervals. At the extremes, we can be 0% confident, in which case our interval is just the point estimate \(\overline{x}\text{.}\) We can also be 100% confident, in which case our confidence interval extends infinitely in both directions. Neither of these is practical, so we settle for “statistically significant” confidence levels such as 95% and 99%.

We can also construct upper or lower confidence bound. These one-sided confidence intervals work in much the same way as the two-tailed confidence interval above, except that we use \(z_\alpha\) instead of \(z_{\alpha/2}\) for our critical value.

Subsection 4.2.4 Sample Size

In many instances we may want to design our sample to get a particular width for our confidence interval. That is, we decide ahead of time what we want the margin of error to be, and then we figure out how big our sample needs to be to get us that margin of error. To do this, we need to work with the formula for the margin of error. Suppose we want our margin of error to be no more than some upper bound \(E\text{.}\) That is,

To solve this inequality for \(n\text{,}\) the sample size, we start by squaring both sides. This gives:

Then multiplying both sides by \(n\) and dividing by \(E^2\) gives us the final inequality:

Again, we don't usually know the value of \(\sigma\) before we take a sample. To get around this, we can either use a value of \(\sigma\) from some previous study, or we can estimate the standard deviation.

To see this in action, consider the following example.

In the above example, it would probably be a good idea to use \(n = 30\) as the sample size, since then we can apply the central limit theorem and treat this as a normal distribution. We can always use a larger sample that this formula indicates. That will only lead to better results.