Section 8.3 The Sampling Distribution of the Sample Proportion

The Sample Proportion The objective of many statistical investigations is to draw a conclusion about the proportion of individuals or objects in a population that possess a specified property. One that has the property of interest is labeled a success (s), and the ones that do not are labeled a failure (f). For example, what proportion of students are wearing jeans could be answered using the sample proportion.

Properties of the Sample Proportion The letter π denotes the proportion of S’s in the population. It is a number between 0 and 1 (times 100 for %). The letter p is the sample proportion of S’s. p = the number of S’s in the sample n

Properties of the Sampling Distribution of p Let p be the proportion of successes in a random sample of size n from a population whose proportion of S’s (successes) is π. Denote the mean of p by μ p and the standard deviation by σ p. Then the following rules hold: –Rule 1: μ p =  –Rule 2: σ p = –Rule 3: When n is large and  is not too near 1 or 0, the sampling distribution of p is approximately normal.

Conditions for Using Rule #3 The farther the value of  is from.5, the larger n must be for a normal approximation to the sampling distribution of p to be accurate. A conservative rule of thumb is that if both n  > 10 and n(1-  ) >10, then it is safe to use a normal approximation.

Ex: The article “Unmarried Couples More Likely to Be Interracial” (San Luis Obispo Tribune, March 13, 2002) reported that for unmarried couples living together, the proportion that are racially or ethnically mixed is 7% A random sample of n = 100 couples will be selected from this population and p, the proportion of unmarried couples that are mixed racially or ethnically will be computed. Compute the mean and the standard deviation of the sampling distribution of p.

Ex: The article “Unmarried Couples More Likely to Be Interracial” (San Luis Obispo Tribune, March 13, 2002) reported that for unmarried couples living together, the proportion that are racially or ethnically mixed is 7%. A random sample of n = 100 couples will be selected from this population and p, the proportion of unmarried couples that are mixed racially or ethnically will be computed. Is it reasonable to assume that the sampling distribution of p is approximately normal for random samples of size n = 100? Explain.

Ex: The article “Unmarried Couples More Likely to Be Interracial” (San Luis Obispo Tribune, March 13, 2002) reported that for unmarried couples living together, the proportion that are racially or ethnically mixed is 7%. Suppose that the sample size is n = 200 rather than n = 100. Does the change in sample size change the mean and standard deviation of the sampling distribution of p? If so, what are the new values for the mean and standard deviation? If not, explain why not.

Ex: The article “Unmarried Couples More Likely to Be Interracial” (San Luis Obispo Tribune, March 13, 2002) reported that for unmarried couples living together, the proportion that are racially or ethnically mixed is 7%. Suppose that the sample size is n = 200 rather than n = 100. Is it reasonable to assume that the sampling distribution of p is approximately normal for random samples of size n = 200? Explain.

Ex: The article “Unmarried Couples More Likely to Be Interracial” (San Luis Obispo Tribune, March 13, 2002) reported that for unmarried couples living together, the proportion that are racially or ethnically mixed is 7%. When n = 200, what is the probability that the proportion of unmarried couples in the sample who are racially or ethnically mixed will be greater than.10?

Assignment p. 427; 27ace, 28-30, Quiz next week on Thursday Complete any reading missed this week.