Confidence Intervals Inferences about Population Means and Proportions

Lecture Objectives You should be able to: 1.Define Key terms including Confidence Level, Margin of Error, Standard Error of Means 2.Compute confidence intervals for population means and proportions 3.Explain in your own words the connection between the Central Limit Theorem and the computation of confidence intervals.

Central Limit Theorem Regardless of the population distribution, the distribution of the sample means is approximately normal for sufficiently large sample sizes (n>=30), with and

Applying CLT If the sample means are normally distributed, what proportion of them are within ± 1 Standard Error? what proportion of them are within ± 2 Standard Errors? If you take just one sample from a population, how likely is it that its mean will be within 2 SEs of the population mean? How likely is it that the population mean is within 2 SEs of your sample mean?

The population mean is within 2 SEs of the sample mean, 95% of the time. Thus, is in the range defined by: 2*SE, about 95% of the time. (2 *SE) is also called the Margin of Error (MOE). 95% is called the confidence level. Confidence Intervals

The Standard Normal Distribution 68% 95% 99.7%

Confidence Interval for Mean In general, the confidence interval for is given by z. is the sample mean z is the confidence factor. It is the number of standard errors one has to go from the mean in order to include a certain percent of observations. For 95% confidence the value is 1.96 (approximately 2.00). is the standard error of the sample means.

Confidence Interval for Mean [2] Since is generally not known we substitute the sample standard deviation, ‘s’. This changes the distribution of the sample means from z (standard normal) to a t-distribution, a close relative. t. The t value is slightly larger than the z for a given confidence level, thereby increasing the margin of error. That is the price of using s in place of

Example – CI for Mean A sample of 49 gas stations nationwide shows average price of unleaded is $ 3.87 and a standard deviation of $ Estimate the mean price of gas nationwide with 95% confidence. In Excel, compute t with 5% error and (n-1), or 48 degrees of freedom =tinv(0.05,48) = , rounded to % CI for the Mean is: t =3.87 ± [2.01 * (0.15/ √ 49)] = $ 3.87 ± Thus, $3.827 < < $3.913 Interpret the result!

CI for Proportions For proportions, p = population proportion = sample proportion Confidence Interval for p is given by ± z. Note similarity to formula for Means. What is the SE here?

Example - Proportions The Wall Street Journal for Sept 10, 2008 reports that a poll of 860 people shows a 46% support for Sen. Obama as President. Find the 95% CI for the proportion of the population that supports him. 95% CI is: = 0.46 ± Thus,.427 < p <.493

How big a sample? A TV station hires you to conduct an opinion poll. They want to know the proportion of the U.S. population that believes that alien landings on Earth have occurred. You are to ensure that the margin of error in the inference from the sample is as close to 3% as possible, for a 95% confidence interval. How big a sample do you need? [sample size.xls].sample size.xls