7.1: Point Estimates vs. Interval Estimates
- Page ID
- 58915
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Imagine you’re surveying your fellow students to understand how much sleep they 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? Not necessarily; your sample of 35 students may have randomly selected more students who sleep less than average, giving you an underestimate of the true average.
There is no way to know this however, so it is better to report a range of values such as “We estimate the average is somewhere between 6.0 and 6.6 hours.” There is a methodology to calculating this range of values that guarantees a probability of successfully including the true value in our interval.
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.
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: An interval may suggest a proportion of 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: An interval might be 131.8 to 144.2 gallons/day.
In this example, we could also construct an interval estimate for the standard deviation.
Related Video
Looking Ahead
In the next section, we’ll introduce confidence intervals where we combine sample statistics and the standard error to create an interval estimate that has a specific level of confidence (like 95%).
This lets us say: “We’re 95% confident that the true value lies in this range.”