9.2: Sample Proportions

Introduction What proportion of U.S. teens know that 1492 was the year in which Columbus “discovered” America? A Gallop Poll found that 210 of 501 teens knew: 0.42 is the statistic we use to gain information about the unknown population parameter p.

Introduction

Do You Remember? Considering binomial distributions… Multiplying each term in a distribution by the same constant will have what effect on the mean and standard deviation? The mean and standard deviation will be multiplied by the same constant.

Do You Remember? A binomial distribution can be reasonably approximated by a Normal distribution as long as the sample size is small relative to the population. We can use the following constraints:

Sampling Distribution of a Sample Proportion How good is a statistic as an estimate of the parameter? To find out, we ask, “What would happen if we took many samples?” The sampling distribution of answers this question.

Sampling Distribution of a Sample Proportion The mean of the sampling distribution of is exactly p. The standard deviation of the sampling distribution of is You will see how these formulas are derived in your reading…

More on Standard Deviation The standard deviation of the sampling distribution of is Because n appears in the denominator, the standard deviation gets smaller as n increases. In order to cut the standard deviation in half, we must take a sample that is four times as large.

Rule of Thumb 1 The formula for standard deviation doesn’t apply when the sample is a large part of the population. Rule of Thumb 1 Use recipe of s.d. only when the population is at least 10 times as large as the sample (N>10n)

Rule of Thumb 2 We will use the Normal approximation to the sampling distribution of for values of n and p that satisfy np>10 n(1 - p)>10

Ex 1: Applying to College A polling organization asks an SRS of 1500 first-year college students whether they applied for admission to any other college. 35% of all first-year 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?

Ex 1: Applying to College Rule of Thumb 1 N > 10n 1.7 million first-year students > (10)(1500) Rule of Thumb 2 np > 10 (1500)(0.35) > 10 n(1 – p) > (1 – 0.35) > 10

Ex 1: Applying to College Because the Rules of Thumb have been satisfied, we will get accurate figures using the standard deviation and Normal approximation. Thus, we want to find the probability that falls between 0.33 and 0.37.

Ex 1: Applying to College Almost 90% of all samples will give a result within 2 percentage points of the truth about the population. We say… “the probability that the value of p-hat is between 0.33 and 0.37”

Ex 2: Survey Undercoverage? One way of checking the effect of undercoverage or nonresponse errors is to compare the sample with known facts about the population. About 11% of U.S. citizens are African American. The proportion of blacks in an SRS of 1500 adults should be close to If only 9.2% of a national sample is represented by blacks, should we suspect under- representation? We will find the probability that a sample contains no more than 9.2% of black citizens.

Ex 2: Survey Undercoverage? Rule of Thumb 1 N > 10n Approximately 30 million > (10)(1500) Rule of Thumb 2 np > 10 (1500)(0.11) > 10 n(1 – p) > (1 – 0.11) > 10

Ex 2: Survey Undercoverage? Only 1.29% of all samples would have such a low percentage. Therefore, we have good reason to suspect under- representation.

Summary Select a large SRS from a population. The sampling distribution of successes is approximately Normal. And…