Estimating a Population Proportion Section 7-2

Binomial Trials Suppose that we conduct 1,000 binomial trials with p = 0.32 We know that for the distribution: μ = np = 1000(0.32) = 320

Binomial Trials We also know that the distribution of the number of "successes" will be approximately normal. Furthermore, we can calculate that P(290 < x < 350) ≈ 0.9546 Now consider the proportion of "successes" in 1,000 binomial trials.

Success Rate

More Binomial Trials Now suppose that we conduct 1,000 binomial trials where we do not know the value of p. Specifically, suppose that a coin is tossed 1,000 times and we observe that it lands heads up 674 times. The law of large numbers tells us that the more the coin is tossed, the closer to the true probability of "success" the relative frequency probability becomes.

How Close? We do not know exactly how close the relative frequency approximation is to the true probability of "success", but we can construct an interval, centered at p-hat, that is likely to contain the true probability of "success." This interval is known as a confidence interval.

Constructing the Interval Before we construct the confidence interval, we need to answer the question of "how confident do we want to be?" The more confident we wish to be that the interval contains the probability of "success", the larger the interval will become. The confidence level is the probability, denoted as 1 - α, that the confidence interval contains the population parameter.

Another Greek Letter? α(alpha) is used here to represent the area in two tails of a standard normal density curve. It relates to zα/2 (called a critical value), which is a value from a standard normal distribution. zα/2 is defined such that: P(Z > zα/2 ) = α/2 where Z is a random variable from a standard normal distribution. Table A-2 has a small chart which contains some common critical values.

Margin of Error The margin of error, E, is the maximum likely difference between p-hat and p (with a probability of 1 - α). For a given confidence level, (such as 0.95) we can find the value of zα/2 and then find the margin of error using the formula:

The form of a Confidence Interval Confidence intervals for a proportion are of the form: (p-hat - E, p-hat + E) where

95% Confidence Interval So it follows that n = 1,000 and To construct a 95% confidence interval for the coin toss data, recall that the coin landed heads up 674 times in 1,000 tosses. So it follows that n = 1,000 and p-hat = 674/1000 = 0.674. From table A2, we find that zα/2 = 1.96

The Interval p-hat = 0.674 and E = 0.0291 p-hat - E = 0.674 - 0.0291 = 0.6449 p-hat + E = 0.674 + 0.0291 = 0.7031 So the 95% confidence interval for the proportion is: (0.6449, 0.7031)

Confidence Intervals and TI Calculators It should come as no surprise that the TI-83 and TI-84 calculators have a function that can be used to construct confidence intervals. The user needs to enter the number of trials (n), the number of "successes" (x) and the confidence level (1 - α). Using the TI-83/TI-84 calculator to construct confidence intervals will be explained in a PowerPoint demonstration.

Interpreting Confidence Intervals The 95% confidence interval based on 1000 binomial trials and 674 successes is (0.6449, 0.7031) The correct interpretation of this confidence interval is "we are 95% confident that the interval actually does contain the true value of the population proportion"

Avoiding Incorrect Interpretations One incorrect interpretation of the confidence interval is "There is a 95% chance that the value of p will fall between 0.6449 and 0.7031" The reason that this is incorrect is that the value of p is fixed, it is not a random variable.

Avoiding Incorrect Interpretations Another incorrect interpretation of the confidence interval is "95% of sample proportions will fall between 0.6449 and 0.7031" The reason that this is incorrect is that we are uncertain what the true value of p is. If the p-hat is an outlier then less than 95% of sample proportions will fall in the interval.

Determining Sample Size The results of a sample are often used to make an educated guess about a population parameter. One example is when supporters of a political candidate or a ballot referendum conduct a poll to determine how much support they can expect from the voters.

Determining Sample Size Assuming that a sample represents a cross section of the population, the pollsters still need to determine how many people to survey. The three things that determine the minimum sample size are: 1. What is an acceptable margin of error? 2. How confident do we wish to be? 3. Is there any prior data to estimate p-hat?

Sample Size Formulas If there is prior knowledge to estimate p-hat then the formula for the minimum sample size is

Sample Size Formulas If there is no prior knowledge to estimate p-hat then the formula for the minimum sample size is

A New Mayor? Jessica Hill is considering running for mayor of Metropolis. She wishes to determine what proportion of people support her campaign and wants to be 95% confident that the margin of error of the result is at most 3% (or 0.03). How many people should her campaign volunteers survey?

A New Mayor? The acceptable margin of error is E = 0.03 Since the confidence level is 95%, this means that zα/2 = 1.96 Jessica is a new candidate, so there is no prior knowledge (no job approval rates, no prior poll results etc.) so use the second formula.

A New Mayor?

Rounding the Result When determining minimum sample sizes, the rule for rounding is that if the formula does not give an integer, round up to the next integer. Since 1067.111 is not an integer then round up to 1068. This means that Jessica Hill's campaign staff should survey a minimum of 1068 people.

Re-Elect Mayor Hill? After winning office, Jessica Hill is running for re- election. This of course means that her staff will be conducting another poll to determine how much support she has. Suppose that her last approval rating was at 63%. Furthermore, suppose that she wishes to be 95% confident that the poll results are within 2% of her true level of support among likely voters. How many people should her campaign staff survey?

Finding the Sample Sizes Since we wish to be 95% confident, this means that zα/2 = 1.96. The prior approval rating of 63% means that p-hat = 0.63. E = 0.02. Since we have a value of p-hat, the first formula can be used.

Sample Size Calculations

Rounding the Result Since 2238.6924 is not an integer then round up to 2239 This means that Jessica Hill's campaign staff should survey a minimum of 2239 people to determine her level of support for the next election.

Homework Section 7-2 (page 337) 5, 7, 9, 11, 13, 15 (Just find Confidence Interval), 17(b), 21(b), 23(b), 27(a), 29, 31