Elementary Statistics Confidence Intervals 6 Elementary Statistics

Confidence Intervals for the Mean Section 6.1 Confidence Intervals for the Mean (large samples)

Point Estimate DEFINITION: A point estimate is a single value estimate for a population parameter. The best point estimate of the population mean is the sample mean An unbiased estimator is just as likely to overestimate as to underestimate the parameter it is estimating. The sample mean is an unbiased estimator of the population mean. From chapter 5 students know that the mean of the sample means is the population mean.

Example: Point Estimate A random sample of 35 airfare prices (in dollars) for a one-way ticket from Atlanta to Chicago. Find a point estimate for the population mean, . 99 101 107 102 109 98 105 103 101 105 98 107 104 96 105 95 98 94 100 104 111 114 87 104 108 101 87 103 106 117 94 103 101 105 90 The sample mean is Sometimes students will work from raw data and other times summary statistics. The raw data are presented here but summary statistics are given in the interest of time. After raw data is given, ask students what single number could be used to estimate the population mean. The point estimate for the price of all one way tickets from Atlanta to Chicago is $101.77.

Interval Estimates • • Point estimate 101.77 An interval estimate is an interval or range of values used to estimate a population parameter. ( ) • 101.77 The chances the the archer hits the exact point center are virtually 0. The probability he hits within the center ring is his level of confidence for that ring. Discuss as the rings widen, the level of confidence goes up but accuracy is sacrificed. The level of confidence, x, is the probability that the interval estimate contains the population parameter.

Distribution of Sample Means When the sample size is at least 30, the sampling distribution for is normal. Sampling distribution of For c = 0.95 0.95 Have students calculate other critical numbers for other values of c. For c = 90%, z.90= 1.645 For c = 99%, z.99 =1.575 0.025 0.025 z -1.96 1.96 95% of all sample means will have standard scores between z = -1.96 and z = 1.96

Maximum Error of Estimate The maximum error of estimate E is the greatest possible distance between the point estimate and the value of the parameter it is, estimating for a given level of confidence, c. When n > 30, the sample standard deviation, s, can be used for . Find E, the maximum error of estimate for the one-way plane fare from Atlanta to Chicago for a 95% level of confidence given s = 6.69. Calculate the error of estimate for c = .90 and c= .95. Have students compare the widths of the intervals at various levels of confidence. Using zc = 1.96, s = 6.69, and n = 35, You are 95% confident that the maximum error of estimate is $2.22.

Confidence Intervals for Definition: A c-confidence interval for the population mean is Find the 95% confidence interval for the one-way plane fare from Atlanta to Chicago. You found = 101.77 and E = 2.22 Left endpoint Right endpoint Encourage students to use a number line to form confidence intervals. • 101.77 ( ) 103.99 99.55 With 95% confidence, you can say the mean one-way fare from Atlanta to Chicago is between $99.55 and $103.99.

Sample Size Given a c-confidence level and an maximum error of estimate, E, the minimum sample size n, needed to estimate , the population mean is You want to estimate the mean one-way fare from Atlanta to Chicago. How many fares must be included in your sample if you want to be 95% confident that the sample mean is within $2 of the population mean? Rework the problem with different values of E. Compare. What effect does increasing E, the maximum tolerance have on the minimum required sample? You should include at least 43 fares in your sample. Since you already have 35, you need 8 more.

Confidence Intervals for the Mean Section 6.2 Confidence Intervals for the Mean (small samples)

The t-Distribution If the distribution of a random variable x is normal and n < 30, then the sampling distribution of is a t-distribution with n – 1 degrees of freedom. Sampling distribution n = 13 d.f. = 12 c = 90% .90 .05 .05 Discuss degrees of freedom. Find other critical numbers for different sample sizes. Show other t-distributions. If the number of degrees of freedom is 30 or more, the t-distributions are very close to the standard normal distribution. t -1.782 1.782 The critical value for t is 1.782. 90% of the sample means (n = 13) will lie between t = -1.782 and t = 1.782.

Confidence Interval–Small Sample Maximum error of estimate In a random sample of 13 American adults, the mean waste recycled per person per day was 4.3 pounds and the standard deviation was 0.3 pound. Assume the variable is normally distributed and construct a 90% confidence interval for . 1. The point estimate is = 4.3 pounds The process for forming a confidence interval is virtually the same as that for large samples. Note though that the population standard deviation cannot be estimated by s. 2. The maximum error of estimate is

Confidence Interval–Small Sample 1. The point estimate is = 4.3 pounds 2. The maximum error of estimate is Right endpoint Left endpoint The process for forming a confidence interval is virtually the same as that for large samples. Note though that the population standard deviation cannot be estimated by s. • 4.3 ( ) 4.152 4.448 4.15 < < 4.45 With 90% confidence, you can say the mean waste recycled per person per day is between 4.15 and 4.45 pounds.