Math 115 Chapter 7-1 Estimating a Population Proportion

Critical Values

A critical value is the number on the borderline separating sample statistics that are significantly high or low from those that are not significant. The number z(alpha/2) is a critical value that is a z score with the property that it is at the border that separates an area of alpha/2 in the right tail of the standard normal distribution

Confidence Interval

A confidence interval (or interval estimate) is a range (or an interval) of values used to estimate the true value of a population parameter. A confidence interval is sometimes abbreviated as CI

The Process Success Rate

A confidence level of 95% tells us that the process we are using should, in the long run, result in confidence interval limits that contain the true population proportion 95% of the time

Confidence Interval for Estimating a Population proportion p: Notation

p = population proportion p-hat = sample proportion n = number of sample values E = margin of error z(alpha/2) = critical value: the z score separating an area of alpha/2 in the right tail of the standard normal distribution

Determining Sample Size: Finding the Sample Size Required to Estimate a Population Proportion: Notation

p = population proportion p-hat = sample proportion n = number of sample values E = margin of error z(alpha/2) = critical value: the z score separating an area of alpha/2 in the right tail of the standard normal distribution

Confidence Interval for Estimating a Population Proportion p: Confidence Interval Estimate of p

p-hat - E < p < p-hat + E where E = z(alpha/2)sqrt(p-hat*q-hat/n) The confidence interval is often expressed in the following formats: p-hat +- E or (p-hat - E, p-hat + E)

Confidence Interval for Estimating a Population proportion p: Requirements

1. The sample is a simple random sample 2. The conditions for the binomial distribution are satisfied: There is a fixed number of trials, the trials are independent, there are two categories of outcomes, and the probabilities remain constant for each trail 3. There are at least 5 successes and at least 5 failures

Procedure for Constructing a Confidence Interval for p

1. Verify that the requirements in the preceding slides are satisfied 2. Use technology or Table A-2 to find the critical value z(alpha/2) that corresponds to the desired confidence level 3. Evaluate the margin of error E = z(alpha/2)sqrt(p-hat*q-hat/2) 4. Using the value of the calculated margin of error E and the value of the sample proportion p-hat, find the values of the confidence interval limits p-hat - E and p-hat + E. Substitute those values in the general format for the confidence interval 5. Round the resulting confidence interval limits to three significant digits

Margin of Error for Proportions

Formula E = z(alpha/2)sqrt(p-hat*q-hat/n

Better Performing Confidence Intervals

Plus Four Method The plus four confidence interval performs between than the Wald confidence interval in the sense that its coverage probability is closer to the confidence level that is used The plus four confidence interval uses this very simple procedure: Add 2 to the number of successes x, add 2 to the number of failures (so that the number of trials n is increased by 4), and then find the Wald confidence interval as described in Part 1 of this section

Finding the Point Estimate and E from a Confidence Interval

Point estimate of p: p-hat = (upper confidence interval limit) + (lower confidence interval limit) / 2 Margin of error: E = (upper confidence interval limit) - (lower confidence interval limit) / 2

Clopper Pearson Method

The Clopper_pearson method is an "exact" method in the sense that it is based on the exact binomial distribution instead of an approximation of a distribution. It is criticized for being too conservative in this sense: When we select a specific vonfidence level, the

Confidence Level

The confidence level is the probability 1 - alpha (such as 0.95, or 95%) that the confidence interval actually does contain the population parameter, assuming that the estimation process is repeated a large number of times. (The confidence level is also called the degree of confidence, or the confidence coefficient)

Determining Sample Size: Finding the Sample Size Required to Estimate a Population Proportion: Requirements

The sample must be a simple random sample of independent sample units When an estimate p-hat in known: n = [z(alpha/2)]^2 * p-hat*q-hat / E^2 When no estimate p-hat is known: n = [z(alpha/2)]^2 * 0.25 / E^2

Analyzing Polls

When analyzing results from polls, consider the following: 1. The sample should be a simple random sample, not an inappropriate sample 2. The confidence level should be provided 3. The sample size should be provided 4. Except for relatively rare cases, the quality of the poll results depends on the sampling method and the size of the sample, but the size of the population is usually not a factor

Margin of Error

When data from a simple random sample are used to estimate a population proportion p, the difference between the sample proportion p-hat and the population proportion p is an error. The maximum likely amount of that error is the margin of error, denoted by E. There is a probability p-hat and p is E or less. The margin of error E is also called the maximum error of the estimate and can be found by multiplying the critical value and the estimated standard deviation of sample proportions

Point Estimate

a point estimate is a single value used to estimate a population parameter. The sample proportion p-hat is the best point estimate of the population proportion p Unbiased Estimator We use p-hat as the point estimate of p because it is unbiased and it is the most consistent of the estimators that could be used An unbiased estimator is a statistic that targets the value of the corresponding population parameter in the sense that the sampling distribution of the statistic has a mean that is equal to the corresponding population parameter The statistic p-hat targets the population proportion p