Chapter 11: Estimation of Population Means

We’ll examine two types of estimates: point estimates and interval estimates.

Point Estimates A point estimate is an estimate of a population parameter that is stated as a single value. An sample statistic will be used as a point estimate for a population parameter. The sampling distribution of sample means and the sampling distribution of sample proportions typically follow the Normal distribution for sufficiently large samples.

Interval Estimates An interval estimate is an estimate of a population parameter that is stated as a range of values within which the population value is likely to fall. Since it is a range, it is defined by two numbers. Interval estimates are also known as confidence intervals. The benefit of using a confidence interval over a point estimate is that confidence intervals provide a better chance of being correct. Think about if you tried predicting tomorrow’s weather. Are you more likely to be correct if you say exactly 86 degrees, or between 80 and 92 degrees?

The degree of confidence associated with a confidence interval details what proportion of interval estimates would correctly contain the population parameter. Common choices are 90%, 95%, and 99%. These levels correspond to percentages of the area of the Normal curve. For example, a 95% confidence interval covers 95% of the Normal curve, leaving 5% area outside the desired region. The area outside the desired region represents the probability or chance of being wrong and is referred to as the level of significance, or alpha level

The level of significance is the maximum amount of error the researcher is willing to risk in case of making an incorrect statement. For example, for a 95% confidence interval, there is a level of significance of 5%. This means that if an infinite number of samples was drawn from the population and confidence intervals were constructed, 95% of the confidence intervals would correctly contain the population parameter, and 5% would not contain the parameter. Because the Normal curve is symmetric, the level of significance has to get split in each tail. So a 5% level of significance would result in 2.5% in each tail.

Confidence intervals will be calculated by taking the single-value estimate and adding and subtracting a margin of error would be our lower limit, and 24.5 would be our upper limit.

When we know the population standard deviation (or we can reasonably estimate it from a similar study that was done in the past) and our sample size is large (30 or more) we use the Z distribution to calculate confidence intervals.

Notice what happens to the range of values as your confidence increases.

One of the problems with the z-distribution is that it assumes we know the population standard deviation, which more often than not is unlikely. So what do we do when we don’t know the population standard deviation, or our sample size is smaller than 30? We have to utilize the t-distribution. The t-distribution is used for small samples, or when the population standard deviation is unknown, so it is estimated with a sample standard deviation. The t-distribution is similar to the z-distribution in that it is bell shaped, symmetrical, and has a mean of 0. However, the t-distribution is more variable than the z-distribution because there is a different value for t for each possible sample size of n.

Using the table, you find the degrees of freedom on the left-hand side. You then slide over to the right based on alpha. – For example, if we were constructing a 95% confidence interval for a sample size of 9, we would go to 9 on the left-hand side, and our alpha would be 5% (.05), which would result in in each tail. Notice how these two columns align. So we would go over to Area in two tails 0.05 and notice that our t-value is Confidence interval for the mean using t:

Df=n-1=16-1=15 5% alpha t=2.131

Df=25-1=24.10 alpha t=1.711

Back to Proportions

To find a confidence interval for a population proportion:

Example: A food company is planning to market a new kind of cereal. However, before marketing the product, the company wants to find what percentage of people will like it. The company’s research department selected a random sample of 500 people and asked them to taste the cereal. Of these 500 people, 290 said they liked it. Construct a 99% confidence interval to find the percentage of all people that will like the cereal. We’re 99% confident that the true proportion of people that like this cereal will be between 52.3% and 63.7%.

Example: The creators of “Meow That’s What I Call Music, Volume 3” decided to test their new CD out on a sample of 2823 cats. Of these cats, 400 liked the CD. Construct a 95% confidence interval to estimate the proportion of the cat population that will like this CD. We are 95% confident that between 12.9% to 15.5% of the cat population will enjoy this CD.