CHAPTER 9 Sampling Distributions 9.2 Sample Proportions

Sample Proportions FIND the mean and standard deviation of the sampling distribution of a sample proportion. CHECK the 10% condition before calculating the standard deviation of the sample proportions. DETERMINE if the sampling distribution of sample proportions is approximately Normal. If appropriate, use a Normal distribution to CALCULATE probabilities involving a sample proportion.

Proportions – Review Symbols/Vocab 𝑝 =

The Sampling Distribution of Consider the approximate sampling distributions generated by a simulation in which SRSs of Reese’s Pieces are drawn from a population whose proportion of orange candies is 0.15. What do you notice about the shape, center, and spread of each? Population n=5 n=20 n=50 Shape Unknown Center Spread

The Sampling Distribution of What did you notice about the shape, center, and spread of each sampling distribution? These are general observations. We will get more specific in a few minutes. Shape: The sampling distribution of 𝑝 can be approximated by a Normal curve in certain circumstances. This depends on the values of n and 𝑝 RULE: As long as np≥10 and n(1-p)≥10, the sampling distribution of 𝑝 can be approximated by a Normal curve

The Sampling Distribution of What did you notice about the shape, center, and spread of each sampling distribution?

The Sampling Distribution of What did you notice about the shape, center, and spread of each sampling distribution? **This formula can only be used when the sample size is no more than 10% of the population size n ≤ (1/10)N**

The Sampling Distribution of In Chapter 8, we learned that the mean and standard deviation of a binomial random variable X are As sample size increases, the spread decreases.

The Sampling Distribution of Sampling Distribution of a Sample Proportion As n increases, the sampling distribution becomes approximately Normal. Before you perform Normal calculations, check that the Normal condition is satisfied: np ≥ 10 and n(1 – p) ≥ 10.

The Sampling Distribution of

Example: Planning for College The superintendent of a large school district wants to know what proportion of middle school students in her district are planning to attend a four-year college or university. Suppose that 80% of all middle school students in her district are planning to attend a four-year college or university. Problem: What is the probability that an SRS of size 125 will give a result within 7 percentage points of the true value? Step 1) Define the parameter: p = Step 2) Determine the mean and standard deviation (10% rule!) and verify that the sampling distribution is approx Normal. 𝝁 𝒑 = Check 10% Rule: 𝝈 𝒑 = Check Normality Condition (np ≥ 10 and n(1 – p) ≥ 10)

Example: Planning for College The superintendent of a large school district wants to know what proportion of middle school students in her district are planning to attend a four-year college or university. Suppose that 80% of all middle school students in her district are planning to attend a four-year college or university. Problem: What is the probability that an SRS of size 125 will give a result within 7 percentage points of the true value? The sampling distribution of p-hat is approximately Normal with 𝝁 𝒑 = 0.8 and 𝝈 𝒑 = 0.036 Step3) Calculations! Step4) Conclusions:

Sample Proportions FIND the mean and standard deviation of the sampling distribution of a sample proportion. CHECK the 10% condition before calculating the standard deviation of the sample proportions. DETERMINE if the sampling distribution of sample proportions is approximately Normal. If appropriate, use a Normal distribution to CALCULATE probabilities involving a sample proportion.