Statistics Confidence Intervals for Population Proportions
Point estimate for the population proportion p is given by the proportion of successes in a sample and is denoted by x pˆ n where x is the number of successes in the sample and n is the number in the sample. The point estimate for the number of failures is qˆ 1 pˆ . The symbols pˆ and qˆ are read as “p hat” and “q hat.” In this section we assume: 1) The sample is a random sample, 2) The conditions for the binomial distribution are satisfied, and 3) The normal distribution can be used to approximate the binomial distribution.
Example. If 1544 are surveyed and 633 respond that the economy is getting worse then 633 pˆ 0.4099740933 1544
Confidence intervals A c-confidence interval for the population proportion p is pˆ E p pˆ E where E zc
ˆˆ pq n
The probability that the confidence interval contains p is c. Earlier we learned that a binomial distribution can be approximated by the normal distribution if np 5 and nq 5 . When npˆ 5 and nqˆ 5 , the sampling distribution for pˆ is approximately normal with a mean of μ pˆ p and a standard error (SE) of ˆˆ pq σ pˆ n
Example. If 6 out of 40 students plan to go to graduate school, the proportion of all students who plan to go to graduate school is estimated as ________. The standard error of this estimate is ________. pˆ 6 / 40 0.15 The population proportion is 0.15 qˆ 1 pˆ 0.85
npˆ 5
yes, 6
nqˆ 5
yes, 34
.15 .85 SE 0.056457949 40 The standard error of estimate is 0.056457949
Example. If 90 out of 360 students plan to go to graduate school, the proportion of all students who plan to go to graduate school is estimated as _______. The standard error of this estimate is _______. pˆ 90 / 360 0.25 The population proportion is 0.25 qˆ 1 pˆ 0.75 .25 .75 SE 0.0228217732 360 The standard error of estimate is 0.0228217732
Guidelines: Constructing a confidence interval for a population proportion 1. Find the point estimate pˆ . 2. Verify that the sampling distribution of pˆ can be approximated by the normal distribution. 3. Identify the level of confidence desired. 4. Find the maximum error of estimate. 5. Find the left and right endpoints and form the confidence interval.
Example. A company that makes a drug to reverse hair loss in males claims that 60% of the people who try their drug will grow new hair. A test on 40 randomly selected males experiencing hair loss produced the following results (Yes: Grew new hair; No: No new hair). Yes No No No Yes No No No Yes No Yes Yes Yes Yes No No Yes No Yes Yes Yes Yes Yes Yes No Yes Yes Yes No No No Yes No No Yes Yes No Yes No Yes 22 yes, 18 no Construct a 95% confidence interval for the proportion of all men experiencing hair loss that will grow new hair by taking this drug.
pˆ .55
qˆ .45
Find E, the margin of error. In a creating a 95% CI, we use 1.9600 as our critical z value. E zc
ˆˆ pq .55 .45 1.96 n 40
E = 0.1541749007 Our 95% confidence interval will be our p hat plus and minus E.
With 95% confidence, the interval (.3958, .7042) will contain the population proportion of all men experiencing hair loss.
With 95% confidence, the interval (.39583, .70417) will contain the population proportion of all men experiencing hair loss. We can always reverse engineer our margin of error.
Find the range of the confidence interval and divide this range by 2. Our margin of error equals .15417.
Example. Most people are right-handed and even the right eye is dominant for most people. Molecular biologists have suggested that late-stage human embryos tend to turn their heads to the right. Russian scientist Vlad Gerberding conjectured that this tendency to turn to the right manifests itself in other ways as well, so he studied kissing couples to see if they tend to lean to the right. He and his colleague, Bobby Johnsonovich observed kissing couples in public places such as malls and airports. Both them need to find a hobby. They found that in a random sample of 124 kissing couples, 80 turned their heads to the right.
a. Calculate the value of the sample statistic. pˆ .6452
qˆ .3548
b. Construct a 90% CI for the population parameter based on this sample.
c. Interpret both this interval and the phrase “90% confident.� I am 90% confident that the interval (.57449, .71584) contains the population proportion of kissing couples who turn their heads to the right.
d. Based on this CI, does it appear to be plausible that 50% of all kissing couples turn to the right? Explain. Absolutely, more than 50% in fact. The interval is entirely above the 50% chance.
e. What would happen to the CI if we quadrupled the sample size and the number of kissing couples?
The range of the CI gets smaller.
Increasing sample size to increase precision One way to increase the precision of the confidence interval without decreasing the level of confidence is to increase the sample size. Given a c-confidence level and a maximum error of estimate E, the minimum sample size n needed to estimate p is 2 zc ˆ ˆ n pq E This formula assumes that you have a preliminary estimate for pˆ and qˆ . If not, use pˆ .50 and qˆ .50
Example. According to a study of the 10 largest US domestic airlines, Southwest Airlines has the lowest proportion of late arrivals, at 0.1577. Suppose you were asked to perform a follow-up study for Southwest Airlines in order to update the estimated proportion of late arrivals. What sample size would you use in order to estimate the population proportion to within an error of ±0.04 with 95% confidence? 2
zc 1.96 ˆ ˆ .1577 .8423 n pq E .04 318.9265347 319
2
Example. A public health survey is to be designed to estimate the proportion of a population having defective vision. How many persons should be examined if the public health commissioner wishes to be 98% certain that the error is below .05. There is no knowledge about the value of p? 2
2
zc 2.33 ˆ ˆ .5 .5 n pq 542.89 543 E .05