Confidence Intervals about the Mean, Population Standard Deviation Unknown (Jump to: Lecture | Video )

Point Estimate
Any statistic that estimates the value of a parameter is called a point estimate.

Figure 1.

We rarely know if our point estimate is correct because it is merely an estimation of the actual value. Because of this discrepancy, we construct confidence intervals to help estimate what the actual value of the unknown population mean is.

Figure 2.

Confidence intervals are a point estimate plus/minus a margin of error. The margin of error is determined by several factors:

1. How confident we want to be with our assessment

2. Population standard deviation

3. How large our sample size is

Figure 3.

Let's say we want to create a 95% confidence interval. That means we have an alpha of 0.05(5%) which is split into two equal tails. This 2.5% refers to the value we look up in the t-table in order to find the t-score we need to plug into the equation.

On the Verbal section of the SAT, a sample of 25 test-takers has a mean of 520 with a standard deviation of 80. Construct a 95% confidence interval about the mean.

Because we are using the t distribution, first we must calculate the degrees of freedom.

df = n - 1 = 25 - 1 = 24

Now, we open our t table (click to open) and look up a two tailed test with alpha = 0.05 and 24 degrees of freedom. We find a t of 2.0639.

Figure 4.

After plugging everything into the equation, we find a lower bound of 486.978 and an upper bound of 553.022.

We are 95% confident that the mean SAT score is between 486.978 and 553.022.

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