Confidence Interval Estimation For statistical inference in decision making: Chapter 9

Objectives Confidence Interval Estimation of the Mean ( σ known) Interpretation of the Confidence Interval Confidence Interval Estimation of the Mean ( σ unknown) Confidence Interval Estimation for the Proportion Determining Sample Size

Statistical Inference

Statistical Inference facilitates decision making.

Via sample data, we can estimate something about our population, such as its average value µ, by using the corresponding sample mean, X-bar.

Recall that µ, the population mean to be estimated, is a parameter, while X-bar, the sample mean, is a statistic.

Point Estimate A point estimate is a statistic taken from a sample and is used to estimate a population parameter. However, a point estimate is only as good as the sample it represents. If other random samples are taken from the population, the point estimates derived from those samples are likely to vary. Because of variation in sample statistics, estimating a population parameter with a confidence interval is often preferable to using a point estimate.

Confidence Interval A confidence interval is a range of values within which it is estimated with some confidence the population parameter lies.

An interval estimate provides more information about a population characteristic than does a point estimate. It provides a confidence level for the estimate. Such interval estimates are called confidence intervals Point Estimate Lower Confidence Limit Width of confidence interval Upper Confidence Limit

Confidence Interval Estimation of Population Mean, μ, when σ is known Assumptions –Population standard deviation σ is known –Population is normally distributed –If population is not normal, use large sample Confidence interval estimate: (where Z is the normal distribution’s critical value for a probability of α/2 in each tail)

The confidence interval formula yields a range (interval) within which we feel with some confidence the population mean is located. It is not certain that the population mean is in the interval unless we have a 100% confidence interval that is infinitely wide, so wide that it is meaningless.

Common levels of confidence intervals used by analysts are 90%, 95%, 98%, and 99%.

Example: Suppose there are 69 U.S. and imported beer brands in the U.S. market. We have collected 2 different samples of 25 brands and gathered information about the price of a 6-pack, the calories, and the percent of alcohol content for each brand. Further, suppose that we know the population standard deviation ( ) of price is $1.45. Here are the samples’ information: Sample A: Mean=$5.20, Std.Dev.=$1.41=S Sample B: Mean=$5.59, Std.Dev.=$1.27=S 1.Perform 95% confidence interval estimates of population mean price using the two samples.

Interpretation of the results from –From sample “A” We are 95% confident that the true mean price is between $4.63 and $5.77. We are 99% confident that the true mean price is between $4.45 and $5.95. –From sample “B” We are 95% confident that the true mean price is between $5.02 and $6.16. (Failed) We are 99% confident that the true mean price is between $4.84 and $6.36. After the fact, I am informing you know that the population mean was $4.96. Which one of the results hold? –Although the true mean may or may not be in this interval, 95% of intervals formed in this manner will contain the true mean.

95% Confidence Interval For 95% confidence, α =.05 and α / 2 =.025. The value of Z.025 is found by looking in the standard normal table under = This area in the table is associated with a Z value of An alternate method: multiply the confidence interval, 95% by ½ (since the distribution is symmetric and the intervals are equal on each side of the population mean. (½) (95%) =.4750 (the area on each side of the mean) has a corresponding Z value of 1.96.

Consider a 95% confidence interval: Z= -1.96Z= 1.96 Point Estimate Lower Confidence Limit Upper Confidence Limit Point Estimate Z μ μlμl μuμu

In other words, of all the possible X-bar values along the horizontal axis of the normal distribution curve, 95% of them should be within a Z score of 1.96 from the mean.

Margin of Error Z [σ / √ n]

Example: A business analyst for cellular telephone company takes a random sample of 85 bills for a recent month and from these bills computes a sample mean of 153 minutes. If the company uses the sample mean of 153 minutes as an estimate for the population mean, then the sample mean is being used as a POINT ESTIMATE. Past history and similar studies indicate that the population standard deviation is 46 minutes. The value of Z is decided by the level of confidence desired. A confidence level of 95% has been selected.

153 + /- 1.96( 46/ √ 85) = ≤ µ ≤ The confidence interval is constructed from the point estimate, 153 minutes, and the margin of error of this estimate, + / minutes. The resulting confidence interval is ≤ µ ≤ The cellular telephone company business analyst is 95% confident that the average length of a call for the population is between and minutes.

Interpreting a Confidence Interval For the previous 95% confidence interval, the following conclusions are valid: I am 95% confident that the average length of a call for the population µ, lies between and minutes. If I repeatedly obtained samples of size 85, then 95% of the resulting confidence intervals would contain µ and 5% would not. QUESTION: Does this confidence interval [ to ] contain µ? ANSWER: I don’t know. All I can say is that this procedure leads to an interval containing µ 95% of the time. I am 95% confident that my estimate of µ [namely 153 minutes] is within 9.78 minutes of the actual value of µ. RECALL: 9.78 is the margin of error.

Confidence interval estimates for five different samples of n=25, taken from a population where µ=368 and σ=15

Be Careful! The following statement is NOT true: “The probability that µ lies between and is.95.” Once you have inserted your sample results into the confidence interval formula, the word PROBABILITY can no longer be used to describe the resulting confidence interval.