1. Sample Proportions Section 9.2
Which of the following best describes the sampling distribution of a mean?
It is the particular distribution in which 𝜇 𝑥 =𝜇 and 𝜎 𝑥 =𝜎.
It is a graphical representation of the means of all possible samples.
It is the distribution of all possible sample means from a given population.
It is the distribution of all possible sample means of a given size.
It is the probability distribution for each possible sample size.
Sample Proportions
Section 9.2

2. Recall If a random variable 𝑿 has a distribution 𝑩(𝒏, 𝒑) then: 𝜇 𝑋 =𝑛𝑝
(This is on your formula packet!)
If a random variable 𝑿 has a distribution 𝑩(𝒏, 𝒑) then:
𝜇 𝑋 =𝑛𝑝
𝜎 𝑋 = 𝑛𝑝 1−𝑝

3. Also recall For 𝑿: 𝜇 𝑋 and 𝜎 𝑋 If 𝒂𝑿: 𝜇 𝑎𝑋 =𝑎 𝜇 𝑋 , and 𝜎 𝑎𝑋 =𝑎 𝜎 𝑋
Multiplying each term in a distribution by the same constant has the effect of multiplying both the mean and standard deviation of that distribution by that constant.
For 𝑿: 𝜇 𝑋 and 𝜎 𝑋
If 𝒂𝑿: 𝜇 𝑎𝑋 =𝑎 𝜇 𝑋 , and 𝜎 𝑎𝑋 =𝑎 𝜎 𝑋
(multiplying)

4. The Sampling Distribution of 𝒑
Sampling Distribution – distribution of all possible samples of same size and from same population.
The mean of the sampling distribution of 𝒑 is exactly 𝒑.
𝜇 𝑝 =𝑝
The standard deviation of the sampling distribution of 𝑝 is:
𝜎 𝑝 = 𝑝 1−𝑝 𝑛
Rule of Thumb #1:
Use this formula for standard deviation of 𝑝 only when the population is at least 10 times as large as the sample. (𝑁≥10𝑛)

5. Examining Standard Deviation of 𝒑
𝜎 𝑝 = 𝑝 1−𝑝 𝑛
Since 𝒏 is in the denominator, what will happen to 𝜎 𝑝 when 𝒏 increases?
𝜎 𝑝 will decrease ( 𝜎 𝑝 is less variable in larger samples)
By what factor would we need to increase n in order to decrease 𝜎 𝑝 by 1 2 ?
4, since = 1 2

6. Using the Normal Approximation for 𝒑
The Binomial distribution has a an approximately Normal shape, but only when sample sizes are large
Rule of Thumb #2
We will use the Normal approximation to the sampling distribution of 𝑝 for values of 𝑛 and 𝑝 that satisfy:
𝑛𝑝≥10 and 𝑛 1−𝑝 ≥10

7. Example – Applying to College
A polling organization asks an SRS of 1500 first-year college students whether they applied for admission to any other college.
We know that 35% of all first-year college students applied to colleges besides the one they are attending.
What is the probability that the random sample of 1500 students will give a result within 2 percentage points of this true value?
How do you think we will solve this problem?
Don’t forget to pay attention to the Rules of Thumb!
We will use Normal approximation.

8. Applying to college (cont)
We know that 𝑛 = 1500, and 𝑝 = 0.35
The sampling distribution of 𝑝 has 𝜇 𝑝 =0.35
In order to use our formula for standard deviation the population must be at least 10 times the size of the sample. Check Rule of Thumb 1
So the population must be at least 10∙1500=15,000 people.
There are over 1.7 million first year college students, so we’re ok
𝜎 𝑝 = 𝑝 1−𝑝 𝑛 =
𝜎 𝑝 =0.0123

9. Applying to college (cont)
Can we use a Normal distribution to approximate the sampling distribution of 𝑝 ? Rule of Thumb 2
𝑛𝑝= =525
𝑛 1−𝑝 = =975
Yes, we can.

10. Normal Approximation
Since we want to find the probability that 𝑝 falls within 2 percentage points of 0.35, what values do we want to calculate?
We want to find the probability that 𝑝 falls between 0.33 and 0.37
0.33≤ 𝑝 ≤0.37
𝑧= 𝑝 − where 𝑝 =0.33 and 𝑝 =0.37
𝑧=−1.63 and 1.63
𝑃 0.33≤ 𝑝 ≤0.37 =𝑃 −1.63≤𝑧≤ SKETCH A NORMAL CURVE!
=0.9484−0.0516=0.8968
About 90% of all samples will give a result within 2 percentage points of the truth about the population.