9.1: Point Estimates

Comment 1

You may feel that since it is so intuitive, you could have figured out point estimation on your own, even without the benefit of an entire course in statistics. Certainly, our intuition tells us that the best estimator for $\mu$ should be $\overline{x}$, and the best estimator for p should be $\hat{p}$.

Probability theory does more than this; it actually gives an explanation (beyond intuition) why $\overline{x}$ and $\hat{p}$ are the good choices as point estimators for $\mu$ and p, respectively. In the Sampling Distributions module of the Probability unit, we learned about the sampling distributions of $\overline{X}$ and found that as long as a sample is taken at random, the distribution of sample means is exactly centered at the value of population mean.

$\overline{X}$ is therefore said to be an unbiased estimator for $\mu$ . Any particular sample mean might turn out to be less than the actual population mean, or it might turn out to be more. But in the long run, such sample means are “on target” in that they will not underestimate any more or less often than they overestimate.

Likewise, we learned that the sampling distribution of the sample proportion, $\hat{p}$, is centered at the population proportion p (as long as the sample is taken at random), thus making $\hat{p}$ an unbiased estimator for p.

As stated in the introduction, probability theory plays an essential role as we establish results for statistical inference. Our assertion above that sample mean and sample proportion are unbiased estimators is the first such instance.

Comment 2

Notice how important the principles of sampling and design are for our above results: if the sample of U.S. adults in (example 2 on the previous page) was not random, but instead included predominantly college students, then .56 would be a biased estimate for p, the proportion of all U.S. adults who believe marijuana should be legalized. If the survey design were flawed, such as loading the question with a reminder about the dangers of marijuana leading to hard drugs, or a reminder about the benefits of marijuana for cancer patients, then .56 would be biased on the low or high side, respectively. Our point estimates are truly unbiased estimates for the population parameter only if the sample is random and the study design is not flawed.

Comment 3

Not only are sample mean and sample proportion on target as long as the samples are random, but their accuracy improves as sample size increases. Again, there are two “layers” here for explaining this.

Intuitively, larger sample sizes give us more information with which to pin down the true nature of the population. We can therefore expect the sample mean and sample proportion obtained from a larger sample to be closer to the population mean and proportion, respectively. In the extreme, when we sample the whole population (which is called a census), the sample mean and sample proportion will exactly coincide with the population mean and population proportion.

There is another layer here that, again, comes from what we learned about the sampling distributions of the sample mean and the sample proportion. Let’s use the sample mean for the explanation.

Recall that the sampling distribution of the sample mean $\overline{X}$ is, as we mentioned before, centered at the population mean $\mu$ and has a standard deviation of $\frac{\sigma }{\sqrt{n}}$. As a result, as the sample size n increases, the sampling distribution of $\overline{X}$ gets less spread out. This means that values of $\overline{X}$ that are based on a larger sample are more likely to be closer to $\mu$ (as the figure below illustrates):

Similarly, since the sampling distribution of $\hat{p}$ is centered at p and has a standard deviation of $\sqrt{\frac{p(1-p)}{n}}$, which decreases as the sample size gets larger, values of $\hat{p}$ are more likely to be closer to p when the sample size is larger.