7.1: Point Estimates vs. Interval Estimates

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Imagine you’re helping a local high school understand how much sleep students are getting each night. You collect a random sample of 35 students and find that the average amount of sleep is 6.3 hours.

Would you say the average for all students is exactly 6.3 hours? Probably not. You’d expect a little variation, right?

So instead of reporting only one number, it’s better to say something like: “We estimate the average is somewhere between 6.0 and 6.6 hours.”

This is the heart of inferential statistics using data from a sample to make estimates or judgments about a whole population. And in this chapter, we’re going to introduce confidence intervals, which let us do that in a structured way.

Definition: Point Estimate

A point estimate is a single value calculated from a sample that is used to estimate a population parameter.

For example: the sample mean, $ \bar{x} $, is a point estimate for the population mean, $ \mu $.

Definition: Interval Estimate

An interval estimate gives a range of values for a population parameter, rather than just one number. The most common example is a confidence interval.

Instead of saying “the average GPA is 2.95,” we might say, “the average GPA is between 2.85 and 3.05.”

Examples: Point and Interval Estimates

Here are a few examples of how both types of estimates show up in real data analysis. Use these to help solidify how the two concepts differ.

Example 1: Proportion of Voters

You survey 600 voters, and 372 say they support Ballot Measure A. That’s a sample proportion of $ \hat{p} = \frac{372}{600} = 0.62 $.

Point estimate: 0.62 (sample proportion)
Interval estimate: From here, we could build a 95% confidence interval for the true proportion of voters who support the measure perhaps from 0.58 to 0.66.

Example 2: Average Daily Water Use

In a study of home water usage, a random sample of 40 households shows an average of 138 gallons per day with a standard deviation of 20 gallons.

Point estimate: 138 gallons (sample mean)
Interval estimate: A full 95% confidence interval (based on this sample) might range from 131.8 to 144.2 gallons/day.

Example 3: Median Rent in College Towns

You're analyzing Craigslist listings in several college towns. In one city, your random sample of 25 listings shows a median rent of \$1,080/month.

Point estimate: \$1,080 is the sample median
Interval estimate: Since median-based intervals are harder to compute directly, we might construct a bootstrap interval using resampling techniques.

Example 4: Sleep Study (continued)

Back to the sleep example if we found the mean was 6.3 hours with a standard deviation of 0.7, and we want to estimate the population average:

Point estimate: 6.3 (sample mean)
Interval estimate: A 95% confidence interval might be 6.1 to 6.5 hours, based on standard error and a t-distribution.

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Example 1: Proportion of Voters

Example 2: Average Daily Water Use

Example 3: Median Rent in College Towns

Example 4: Sleep Study (continued)

Looking Ahead

Definition: Point Estimate

Definition: Interval Estimate

Examples: Point and Interval Estimates

Example 1: Proportion of Voters

Example 2: Average Daily Water Use

Example 3: Median Rent in College Towns

Example 4: Sleep Study (continued)

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Looking Ahead