Sampling Distributions 6 Sampling Distributions Lesson 6.4 The Sampling Distribution of a Sample Proportion

The Sampling Distribution of a Sample Proportion Calculate the mean and standard deviation of the sampling distribution of a sample proportion and interpret the standard deviation. Determine if the sampling distribution of is approximately normal. If appropriate, use a normal distribution to calculate probabilities involving .

The Sampling Distribution of a Sample Proportion The sample proportion is the statistic that we use to estimate the unknown population proportion p. To understand how much varies from p and what values of are likely to happen by chance, we want an understanding of the sampling distribution of the sample proportion . Sampling Distribution of the Sample Proportion The sampling distribution of the sample proportion describes the distribution of values taken by the sample proportion in all possible samples of the same size from the same population.

The Sampling Distribution of a Sample Proportion As with the sampling distribution of the sample count X, there are formulas that describe the center and variability of the sampling distribution of . How to Calculate µp^ and σp^ Suppose that is the proportion of successes in an SRS of size n drawn from a large population with proportion of successes p. Then: The mean of the sampling distribution of is The standard deviation of the sampling distribution of is

The Sampling Distribution of a Sample Proportion Here are some important facts about the mean and standard deviation of the sampling distribution of the sample proportion: The sample proportion is an unbiased estimator of the population proportion p. This is because the mean of the sampling distribution µ is equal to the population proportion p. The standard deviation of the sampling distribution of describes the typical distance between and the population proportion p. The sampling distribution of is less variable for larger samples. This is indicated by the in the denominator of the standard deviation formula. The formula for the standard deviation of the distribution of requires that the observations are independent. In practice, we are safe assuming independence when sampling without replacement as long as the sample size is less than 10% of the population size.

The Sampling Distribution of a Sample Proportion Is the sampling distribution of (the sample proportion of successes) related to the sampling distribution of X (the sample count of successes)? Yes!

The Sampling Distribution of a Sample Proportion Both the sample size and the proportion of successes in the population affect the shape of the sampling distribution of the sample proportion . The shape of the sampling distribution of will be closer to normal when the value of p is closer to 0.5 and the sample size is larger. The Large Counts Condition Suppose is the proportion of successes in a random sample of size n from a population with proportion of successes p. The Large Counts condition says that the distribution of will be approximately normal when np ≥ 10 and n(1 - p) ≥10 When the Large Counts condition is met, we can use a normal distribution to calculate probabilities involving = the proportion of successes in a random sample of size n.

LESSON APP 6.4 What’s that spot on my potato chip? A potato-chip producer and its main supplier agree that each shipment of potatoes must meet certain quality standards. If the producer finds convincing evidence that more than 8% of the potatoes in the entire shipment have “blemishes,” the truck will be sent away to get another load of potatoes from the supplier. Otherwise, the entire truckload will be used to make potato chips. To make the decision, a supervisor will inspect a random sample of potatoes from the shipment. Suppose that the proportion of blemished potatoes in the entire shipment is p = 0.08 and that the supervisor randomly selects n = 500 potatoes for inspection.

LESSON APP 6.4 What’s that spot on my potato chip? Calculate the mean and standard deviation of the sampling distribution of . Interpret the standard deviation. Justify that the sampling distribution of is approximately normal. Calculate the probability that at least 11% of the potatoes in the sample are blemished. Based on your answer to Question 3, what should the supervisor conclude if he selects an SRS of size n =500 and finds = 0.11? Explain.

The Sampling Distribution of a Sample Proportion Calculate the mean and standard deviation of the sampling distribution of a sample proportion and interpret the standard deviation. Determine if the sampling distribution of is approximately normal. If appropriate, use a normal distribution to calculate probabilities involving .