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Wize University Statistics Textbook > Estimating with Confidence Intervals

Confidence Interval for a Proportion

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Confidence Interval for a Proportion

The population proportion pp is the parameter we are trying to estimate using the sample proportion p^\hat{p}, which is the statistic or estimate.

The confidence intervals used to estimate the population proportion pp is

p^±z∗p^q^n\boxed{\displaystyle\hat{p}\pm z^{\ast}\sqrt{ \frac{\hat{p}\hat{q}}{n}}}

where,
  • q^=1−p^\hat q=1-\hat p
  • z∗p^q^n\displaystyle z^{\ast}\sqrt{ \frac{\hat{p}\hat{q}}{n}} is the margin of error.

The confidence level corresponds to its z∗z^{\ast} “multiplier” value:



Watch Out!
Remember: We only use z⋆\bm z^\star for proportions!

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Finding z∗\colorOne{z^{\ast}} using the z-table:





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Finding z∗\colorOne{z^{\ast}} using the t-table:




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Example: Confidence Interval for a Proportion

Based on a random sample of 200 households, 70 of them have at least one pet.

(a) What is the point estimate?

The point estimate is the sample statistic. In this case, it is the sample proportion:

p^=70200=0.35\hat{p}=\frac{70}{200}=0.35


(b) For a 95% confidence interval, what is the margin of error?


z⋅=1.96z^{\cdot}=1.96
z∗p^q^n=1.96(0.35)(0.65)200=0.0661z^{\ast}\sqrt{ \frac{\hat{p}\hat{q}}{n}}=1.96\sqrt{\frac{\left(0.35\right)\left(0.65\right)}{200}}=0.0661

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(c) Construct a 95% confidence interval for households that has pets.

p^±z∗p^q^n\hat p \pm z^{\ast}\sqrt{ \frac{\hat{p}\hat{q}}{n}}
0.35±0.06610.35 \pm 0.0661

[0.2839, 0.4161]\left[0.2839,\ 0.4161\right]

We are 95% confident that between 28.39% and 41.61% of all households have pets.


A random sample of 1000 people in Calgary were asked if they are satisfied with their mobile carriers: “YES or NO?” Results: 535 people said YES.
Construct a 95% confidence interval and interpret it in simple English.
Extra Practice
Confidence Interval
You are given an 99% confidence interval for the proportion of homeowners that still have a home phone →\rightarrow (0.35, 0.49)
1. What is an estimate for the proportion of homeowners having a home phone?
2. What is the standard error of the sample proportion of homeowners owing a home phone?
Confidence Interval for a Proportion
A random sample of 1000 people in Calgary were asked if they are satisfied with their mobile carriers: “YES or NO?” Results: 535 people said YES.
Confidence Interval for a Proportion
A random sample of 1000 people in Calgary were asked if they are satisfied with their mobile carriers: “YES or NO?” Results: 535 people said YES.
Confidence Interval for a Proportion
Based on a random sample of 200 households, 70 of them have at least one pet. Construct a 95% confidence interval for the households that has pets.
Confidence Interval for a Proportion
Parts (i) to (iv) are based on the following information:
In 2006, it was reported that only 40% of students at the university ask questions in lectures. It is believed that this percentage has changed. You randomly sampled 80 students and asked them "Have you ever asked a question in a lecture?" Twenty-four of them responded "Yes". There are 20,386 students at the university.
(iii) What is the p-value?
Population sampling
There are 600 Sephora locations in the United States. Christine randomly selected 40 stores, and then she randomly selected 100 customers from each of those stores. Less than 1% of her sample identified as heterosexual males. Based on her sample, the average spending at Sephora is $45. In addition, 340 customers in her sample revealed that they have VIB Rouge status, which the highest tier status that is reached when a customer spends $1,000 in a calendar year.
Which of the following is/are true? (Check all that applies.)
Confidence Interval for a Proportion
Based on a sample, the average time it takes a customer to eat a burger is 20 minutes with a standard deviation of 8 minutes. Furthermore, 27 customers in the sample ordered the vegetarian burger.
We are 95% confident that between 23.1% and 43.6% of all customers order the vegetarian burger. Construct a 95% confidence interval for the average time it takes all customers to eat a burger.
Confidence Interval for a Proportion
Which of the following is true for a 90% confidence interval for the population proportion.
Confidence intervals
Based on a sample of 800 children in Twin Pines, 82% of them are vaccinated. Interpret a 95% confidence interval [0.788, 0.852]\left[0.788,\ 0.852\right] in plain English.
Confidence Interval
You are given an 99% confidence interval for the proportion of homeowners that still have a home phone →\rightarrow (0.35, 0.49)
1. What is an estimate for the proportion of homeowners having a home phone?
2. What is the standard error of the sample proportion of homeowners owing a home phone?