High School
SAT
ACT
University
MCAT
LSAT
DAT
Wize University Statistics Textbook > Sampling Distribution

Sampling Distribution for a Proportion

Sampling Distribution for a MeanPrevious Section
Finite Population SamplingNext Section
Popular Courses
Find My Course
0:00 / 0:00

Sampling Distribution of Proportions

By the Central LImit Theorem, if the sample size nn is large enough, then the sampling distribution for a proportion p^\hat{p} is approximately normal with mean pp and standard deviation σp^=pqn\large\sigma_{\hat{p}}=\large\sqrt{\frac{pq}{n}} , where,
  • pp is the population proportion (the parameter).
  • q=1−pq=1-p

We can then use the standardization formula for normal distribution to find the z-score, which will help us more easily calculate probabilities!
Z=p^−ppqn\boxed{\large Z=\frac{\hat{p}-p}{\large\sqrt{\frac{\large pq}{\large n}}}}

When a randomly selected sample size nn is drawn from a population, the sample proportion p^\hat{p} is denoted:

p^=Xn\large\hat{p}=\frac{X}{n}

0≤p^≤10\leq\hat{p}\leq1

where X is the number of individuals with a certain characteristic.

When is the Sample Size "Large Enough"?

The sample size is large enough if np ≥ 9   and   n(1−p) ≥ 9\boxed{np\ \geq\ 9~~~\text{and}~~~n(1-p)\ \geq\ 9}


Example

A report from 2008 revealed that 12% of undergraduate students avoid enrolling in weekend courses. Based on a recent survey of 70 undergraduate students, 11 of the say that they avoid enrolling in weekend courses. Is the sample size large enough?

np=(70)(0.12)=8.4<9 → NOnp=\left(70\right)\left(0.12\right)=8.4<9\ \rightarrow\ NO
nq=(70)(1−0.12)=(70)(0.88)=61.6>9 → YESnq=\left(70\right)\left(1-0.12\right)=\left(70\right)\left(0.88\right)=61.6>9\ \rightarrow\ YES

Sample is not large enough.
0:00 / 0:00

Example: Sampling Distribution for a Proportion


Landlord Magazine revealed that 85% of renters pay their rents on time. Based on a random sample of 60 renters, what is the probability that more than 55 of them pay their rents on time?




p^=5560=0.9167\hat{p}=\dfrac{55}{60}=0.9167
z=p^−ppqnz=\dfrac{\hat{p}-p}{\sqrt{\dfrac{pq}{n}}}

z=0.9167−0.85(0.85)(0.15)60=1.45z=\dfrac{0.9167-0.85}{\sqrt{\dfrac{\left(0.85\right)\left(0.15\right)}{60}}}=1.45

Using the z-table:

P(z>1.45)=1−P(z<1.45)=1−0.9265=0.0735P\left(z>1.45\right)=1-P(z<1.45)=1-0.9265=0.0735


Miles claims that 4% of sales get refunded. You randomly select 200 transactions. What is the probability that more than 12 transactions in his sample got refunded?

Extra Practice
Sampling distribution for a proportion
Celeb Magazine mentioned that 78% of young adults aged 18 to 24 has heard of Sarah Carpenter. If a random sample consisted of 802 people, what is the probability that less than 75% of the sample has heard of Sarah Carpenter?
Sampling Distribution for a Proportion
A 2013 census recorded that 30% of the students attending a certain school in Toronto travel more than 30 minutes every day to attend school. Assuming that this proportion is still correct today, and a random sample of 40 students is slected from this school. Which of the following statment is TRUE about the sample distribution of this sample proportion?
Sampling Distribution for a Proportion
At the University of Tobermory, 30% of students have a grade 12 average of at least 75%.
Which of the following are the mean and standard deviation for the sample proportion of all samples of 16 students at this university that have a grade 12 average of at least 75%?
Sampling Distribution for a Proportion
Suppose we surveyed 1000 students at a certain university and found that 30% of them are smokers. By the Central limit theorem, since this population is large, we can conclude that 30% of the entire university population are smokers.
Sampling Distribution for a Proportion
A 2013 census recorded that 30% of the students attending a certain school in Toronto travel more than 30 minutes every day to attend school. Assuming that this proportion is still correct today, and a random sample of 40 students is slected from this school. Which of the following statment is TRUE about the sample distribution of this sample proportion?
Sampling Distribution for a Proportion
Suppose that the true proportion of people who like Game of Thrones is 63%. If a random sample has 500 people, what is the probability of the sample proportion being more than 65%?
Sampling Distribution for a Proportion
Suppose that the true proportion of people who like Game of Thrones is 63%. If a random sample has 500 people, what is the probability of the sample proportion being more than 65%?
Probability on Sample Proportion
20% of drivers report feeling tired behind the wheel when driving at night.
a) In a sample of 150 drivers, what’s the probability that over a quarter of them feel tired behind the wheel?
b) In a sample of 150 drivers, what’s the probability that between 15% and 22% feel tired behind the wheel?
Sampling Distribution for a Proportion
8% of all pizza deliveries at Joe's Brothers Pizza are delivered late. Find the probability that in a sample of 600 deliveries, more than 5.4% of deliveries are late.
Sampling Distribution for a Proportion
A 2013 census recorded that 30% of the students attending a certain school in Toronto travel more than 30 minutes every day to attend school. Assuming that this proportion is still correct today, and a random sample of 40 students is slected from this school. Which of the following statment is TRUE about the sample distribution of this sample proportion?
Sampling distribution for a proportion
The latest Teens and Technology study from Pew Research found that 78 percent of teens aged 12 to 17 now own cell phones. Determine the probability that fewer than 78% percent of teens own a cell phone when the sample size is 802 and the population proportion is 75%.
Sampling distribution for a proportion
We want to see if the proportion of male who know how to swim differ than those of females. Here is a contingency table:
The value of the test-statistic is
Sampling distribution for a proportion
We want to see if the proportion of male who know how to swim differ than those of females. Here is a contingency table:
The estimate of the proportion of all males that can swim is
Sampling Distribution of a Proportion
A sample of 80 adults that drink coffee are asked; where do you usually go to get your morning coffee, Tim Hortons, McDonalds, Starbucks, or other? The results are recorded below:
What percent of those that are female go mostly to other coffee shops?
Sampling Distribution of a Proportion
A sample of 80 adults that drink coffee are asked; where do you usually go to get your morning coffee, Tim Hortons, McDonalds, Starbucks, or other? The results are recorded below:
What percent of those that go usually go to Tim Hortons are female?
Sampling Distribution of a Proportion
A group of people were asked about the type of soda pop drink they had the last time they had BBQ. The data is entered in the following table.
What percent of those that had Pepsi were female?
Sampling Distribution of a Proportion
A group of people were asked about the type of soda pop drink they had the last time they had BBQ. The data is entered in the following table.
What percent of males had Coca-Cola?
Sampling Distribution of a Proportion
A group of people were asked about the type of soda pop drink they had the last time they had BBQ. The data is entered in the following table.
What percent of people had neither Coca-Cola nor Pepsi?
Calculation
A sample of 80 adults that drink coffee are asked; where do you usually go to get your morning coffee, Tim Hortons, McDonalds, Starbucks, or other? The results are recorded below:
What percent of those people go to Tim Hortons or Starbucks?
Sampling Distribution for a Proportion
Refer to table in Question 8.
What proportion of customers who saw the ad made a purchase?
Sampling Distribution for a Proportion
A marketing company surveyed 500 customers at a mall. They asked if they each customer two questions:
1) Did you see a Jugo Boss ad? YES/NO
2) Did you (i) make a purchase at Jugo Boss, (ii) went to the store but only browsed, or (iii) did not go to the store?
Sampling Distribution of Proportions
Miles claims that 4% of sales get refunded. You randomly select 200 transactions. What is the probability that more than 12 transactions in his sample got refunded?
Sampling Distribution for a Proportion
A lot people go to Le Deux, a romantic restaurant, on first dates. Which of the following pertains to a proportion distribution?
Sampling Distribution for a Proportion
Lots of people use Instagram. Which of the following pertains to a proportion distribution?
Sampling Distribution
Landlord Magazine revealed that 85% of renters pay their rents on time. Based on a random sample of 60 renters, what is the probability that more than 55 of them pay their rents on time?
Sampling Distribution for a Proportion
Based on a sample of 1000 people, 53% are in favour of same-sex marriage. Based on another sample of 1000 people, 66% are in favour. Given this, which of the following must be true?
Sampling Distribution for a Proportion
A study on wellbeing and stress of university students is conducted. The researcher drew a random sample of 5 departments on campus. Then he randomly sampled 100 students within each department that was selected and handed them the link to his online survey. Ultimately, only 30% of the sample responded. Amongst those who responded, 120 of them say they feel stressed.
(d) What proportion of students in the sample revealed that they feel stressed?
Sampling Distribution 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.
(ii) Which of the condition(s) are met? (Check all that applies.)
Standardizing: Sampling Distribution
Half of the population can swim. There is a 19.77% probability that that more than p^\hat{p}% of my sample of 50 can swim. Find xx.