8.3 A Population Proportion
- Page ID
- 36503
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Section 8.3 A Population Proportion
Learning Objective:
In this section, you will:
- Apply and interpret point estimates and confidence intervals
- Determine adequate sample sizes needed to estimate population parameters
- Construct and interpret confidence intervals for population proportions
Proportion = Probability = Percent – Example: If 28% of scores are higher than yours, then the probability of a score being higher than yours is 0.28, and the proportion of scores higher than yours is 0.28
Point Estimate – the best estimate for a population proportion, p, is the sample proportion, 𝒑̂.
𝒑̂ = 𝒙 𝒒̂ = 𝟏 − 𝒑̂
𝒏
Margin of Error (E) – maximum difference between the sample proportion and the true value of the population proportion.
Confidence Interval – a range of values used to estimate the true value of a population parameter.
𝒑̂ ± 𝑬 or 𝒑̂– 𝑬 < µ < 𝒑̂ + 𝑬 or (𝒑̂– 𝑬, 𝒑̂ + 𝑬)
Confidence level- the probability 1 – (usually expressed as a percentage) that the confidence interval actually does contain the population parameter, assuming that the estimation process is repeated a large number of times.
Example 1: The 90% confidence interval for the proportion of all students with a GPA over 3.5 is
.0997 < p < .2203.
Interpretation: “We estimate with 90% confident that the true value of the proportion of all students with a GPA over 3.5 is between 0.0997 and .2203.” If we construct similar confidence intervals using sample proportions numerous times, we expect that 90% of those intervals would contain the true population proportion.
Calculating Confidence Intervals for population proportions:
Using the Graphing calculator TI-84: STAT, TESTS, A:1-PROPZINT
1-PropZInt(x, n, CL)
Example 2: When Mendel conducted his famous genetics experiments with peas, one sample of offspring consisted of 428 green peas and 152 yellow peas. Find a 95% confidence interval estimate of the percentage of green peas.
Example 2 (continued): Mendel expected that 75% of the peas would be green. Given that the percentage of green peas in our sample is not 75%, do the results contradict Mendel’s theory? Why or why not?
Example 3: If 230 out of 600 teenagers plan to see the new Hunger Games movie, find a 90% confidence interval estimate for the percentage of all teenagers planning to see the movie.
Example 3 (continued): The movie theater claims that 25% of teenagers are planning to see the movie. Does their claim appear to be correct?
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Calculating the Minimum Sample Size
If we know our desired margin of error, we must have a large enough sample to guarantee the desired
𝟐 [𝒛𝜶/𝟐] 𝒑̂𝒒̂ error. 𝒏 = 𝑬𝟐 , when 𝑝̂ and 𝑞̂ are known
𝟐
[𝒛𝜶/𝟐] 𝟎.𝟐𝟓
If we don’t know 𝑝̂ in advance, we use 0.5 for both 𝑝̂ and 𝑞̂. This gives us 𝒏 = 𝑬𝟐
Always round up to next whole number when determining sample size.
Example 4: Find the minimum sample size needed if the margin of error must be two percentage points, the confidence level is 99%, and the point estimate for the population proportion is 14%. Write an interpretation.
Example 5: A survey was conducted to determine the percentage of car owners who would pay to put nitrogen in their tires (Nitrogen supposedly leaks out at a slower rate than air, which keeps the tire pressure at the ideal level.) How many randomly selected car owners should be surveyed? Assume that we want to be 95% confident that the sample percentage is within 3% of the true percentage of all car owners who would be willing to pay for nitrogen. Write an interpretation.
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For more information and examples see online textbook OpenStax Introductory Statistics pages 460-467.
“Introduction to Statistics” by OpenStax, used is licensed under a Creative Commons Attribution License 4.0 license