Section Estimating a Proportion with Confidence Objectives: 1.To find a confidence interval graphically 2.Understand a confidence interval as consisting of those population proportions for which the result from the sample is reasonably likely 3.To always check the three conditions before constructing a confidence interval 4.To construct a confidence interval using the formula 5.To interpret a confidence interval and the meaning of “confidence” 6.To compute the required sample size for a given margin of error

Section Estimating a Proportion with Confidence General idea: Consider the population of the U.S. Suppose you are interested in the proportion of redheads in the population. Since the proportion of redheads is probably unknown, you will have to estimate it. What should you do? –Take a sample. (The size will depend on how much time and money you have.) –Compute the sample proportion. (The Central Limit Theorem tells you that this estimator is unbiased, and has other “desirable” properties.) This is your best guess. –Are you “sure”? What do you mean by sure? How “sure” do you need to be?

Introduction A Pew Research Center survey found that 55% of singles ages say they aren’t in a committed relationship and are not actively looking. This percentage is based on interviews with 1068 singles. The survey reported a margin of error of 3%. The researchers also say that they are 95% confident that the error in the percentage (55%) is less than 3% either way. That is, they are 95% confident that if they were to ask all young singles in the U.S., between 52% and 58% would report that they aren’t in a committed relationship and are not actively looking. What do they mean by this? Section Estimating a Proportion with Confidence

Reasonably Likely Events Section Estimating a Proportion with Confidence

Reasonably Likely Events and Rare Events Reasonably likely events are those in the middle 95% of the distribution of all possible outcomes. The outcomes in the upper 2.5% and lower 2.5% of the distribution are rare events - they happen, but rarely. Section Estimating a Proportion with Confidence Rare Upper 2.5% Rare Lower 2.5% Reasonably Likely Middle 95%

Example: Reasonably Likely Results from Coin Flips Section Estimating a Proportion with Confidence

Example: Reasonably Likely Results from Coin Flips Section Estimating a Proportion with Confidence

Example: Reasonably Likely Results from Coin Flips Section Estimating a Proportion with Confidence

Introduction, continued. Section Estimating a Proportion with Confidence

The Meaning of a Confidence Interval Suppose you take repeated random samples of size 40 from a population with 60% successes. What proportion of successes would be reasonably likely in your sample? Section Estimating a Proportion with Confidence

The Meaning of a Confidence Interval Section Estimating a Proportion with Confidence Reasonably likely sample proportions for n = 40 p(1 - p)  ME = 1.96  CI = p ± ME [0.808, 0.992] [0.677, 0.923] [0.559, 0.841] [0.449, 0.751] [0.345, 0.655] [0.249, 0.551] [0.159, 0.441] [0.077, 0.323] [0.008, 0.192]

The Meaning of a Confidence Interval Reasonably likely sample proportions for samples of size n = 40 Section Estimating a Proportion with Confidence

The Meaning of a Confidence Interval Suppose that in an experiment, 75%, or 30 out of the 40 trials, resulted in success. Is it plausible that the true proportion is 50%? Is it plausible that the true proportion is 80% What values are plausible for the population proportion? Section Estimating a Proportion with Confidence

The Meaning of a Confidence Interval Plausible population percentages are p = 0.6, p = 0.7, p = 0.8 Section Estimating a Proportion with Confidence

The Meaning of a Confidence Interval Plausible population percentages are p = 0.6, p = 0.7, p = 0.8. The sample proportion 0.75 (represented by the red vertical line) intersects the reasonably likely range of values for p = 0.80 (from to 0.923, represented by the orange line segment). If the population proportion is 0.80, you are reasonably likely to get 30 successes in 40 trials, or 75%. The sample proportion 0.75 (represented by the red vertical line) does not intersect the reasonably likely range of values for p = 0.50 (from to 0.655, represented by the orange line segment). If the population proportion is 0.50, you are not likely to get 30 successes in 40 trials, or 75%. Section Estimating a Proportion with Confidence

The Meaning of a Confidence Interval Plausible population percentages are p = 0.6, p = 0.7, p = 0.8 Section Estimating a Proportion with Confidence

The Meaning of a Confidence Interval Plausible population percentages are from about p = 0.6 to about p = These plausible percentages for the population proportion are called the 95% confidence interval for p. Section Estimating a Proportion with Confidence

A Confidence Interval for a Population Proportion Section Estimating a Proportion with Confidence

A Confidence Interval for a Population Proportion Section Estimating a Proportion with Confidence

A Confidence Interval for a Population Proportion Section Estimating a Proportion with Confidence

A Confidence Interval for a Population Proportion Once again, what is it that we are trying to do? We wish to find out the value of an unknown population parameter - the proportion of successes. The best estimate of the value of the population proportion, based on the Central Limit Theorem, is to take a random sample and compute the sample proportion. (Bigger samples are better, etc.) In some applications, it is useful to consider a range or interval of values, instead of just one. Depending on how “confident” we want or need to be, we can construct a confidence interval - a range of likely values for the population proportion. Section Estimating a Proportion with Confidence

A Confidence Interval for a Population Proportion Section Estimating a Proportion with Confidence

A Confidence Interval for a Population Proportion Section Estimating a Proportion with Confidence

Example: Safety Violations Suppose you have a random sample of 40 buses from a large city and find that 24 buses have a safety violation. Find the 90% confidence interval for the proportion of all buses that have a safety violation. Section Estimating a Proportion with Confidence

Example: Safety Violations Suppose you have a random sample of 40 buses from a large city and find that 24 buses have a safety violation. Find the 90% confidence interval for the proportion of all buses that have a safety violation. Using the TI-83/84: STAT TESTS 1-PropZInt ENTER 1-PropZInt x: 24 n: 40 C-Level:.90 Calculate [ENTER] Section Estimating a Proportion with Confidence

The Capture Rate Sometimes a confidence interval “captures” the true population proportion and sometimes it doesn’t. The capture rate of a method of constructing confidence intervals is the proportion of confidence intervals that contain the population parameter (proportion) in repeated usage of the method. If a polling company uses 95% confidence intervals in a large number of different surveys, the population proportion p should be in 95% of them. Section Estimating a Proportion with Confidence

Margin of Error and Sample Size Section Estimating a Proportion with Confidence

Margin of Error and Sample Size Example: The Effect of Sample Size on the Margin of Error Section Estimating a Proportion with Confidence

Margin of Error and Sample Size Example: The Effect of Sample Size on the Margin of Error Section Estimating a Proportion with Confidence

What Sample Size Should You Use? Section Estimating a Proportion with Confidence

What Sample Size Should You Use? Section Estimating a Proportion with Confidence

What Sample Size Should You Use? Example: What sample size should you use for a survey if you want the margin of error to be at most 3% with 95% confidence but you have no estimate of p? Section Estimating a Proportion with Confidence