CHAPTER 7 Confidence Intervals and Sample Size Sample Student Notes Stat 200 Elementary Statistics Introduction Estimation Section 7 1 Confidence Intervals for Mean Known or n 30 and Sample Size Point Estimate Interval Estimate Three Properties of a good estimator 1 2 3 Confidence Level of an interval estimate of a parameter is the that the interval estimate will contain the parameter A Confidence Interval is a specific interval estimate of a parameter determined by using data obtained from a sample and by using the specific confidence level of the estimate The Critical Value E Formula X z 2 X z 2 n n z 2 196 z 2 2 58 Maximum Error of Estimate Example 1 A sample of 50 people who play the lottery found the mean to be 19 and the standard deviation to be 6 8 Find the best point estimate of the population mean and the 95 confidence interval of the population mean Example 2 A survey of 30 adults found that the mean age of a person s primary vehicle is 5 6 years Assuming the standard deviation of the population is 0 8 years find the best point estimate of the population mean and the 99 confidence interval of the population mean Example 3 A study found the body temperatures of 106 healthy adults The sample mean was 98 2 degrees and the sample standard deviation was 0 62 degrees Find the margin of error E and the 95 confidence interval for Example 4 The following data represent a sample of the assets in millions of dollars of 30 credit unions Find the 90 confidence interval of the mean 12 23 2 27 1 47 21 58 2 89 16 56 12 77 12 24 13 19 73 25 11 59 8 74 7 92 1 24 9 16 1 91 6 69 3 17 4 39 2 17 1 42 14 64 1 06 2 76 40 22 5 01 4 78 2 42 18 13 16 85 Sample Size Example 5 The college president wants to estimate the average age of the students at their college How large a sample is necessary He would like to be 99 confident that the estimate should be accurate within 1 year From a previous study the standard deviation of the ages is known to be 3 years Example 6 Assume that we want to estimate the mean IQ score for the population of statistics professors How many statistics professors must be randomly selected for IQ tests if we want 95 confidence that the sample mean is within 2 IQ points of the population mean Assume that s 15 as is found in the general population Section 7 2 Confidence Intervals for the Mean When is Unknown Characteristics of the t distribution Similar to the standard normal distribution 1 2 3 4 Differs from the standard normal distribution 1 2 3 Degree of Freedom Formula s s X t 2 X t 2 n n Example 1 Find the ta 2 value for a 95 confidence interval when the sample size is 22 Example 2 Ten randomly selected automobiles where stopped and the tread depth of the right front tire was measured The mean was 0 32 inch and the standard deviation was 0 08 inch Find the 95 confidence interval of the population mean depth Assume the variable is approx normally distributive Example 3 The data represent a sample of the number of home fires started by candles for the past several years Find the 99 confidence interval for the mean number of home fires started by candles When to Use the z or t Distribution 7 3 Confidence Intervals and Sample Size for Proportions Proportion Notation p X n X and q or 1 p n n Example 1 In a recent survey of 150 households 54 had central air conditioning Find p and q where p is the proportion of households that have central air conditioning Formula p za 2 p q p p za 2 n p q n Example 2 A sample of 500 nursing applications included 60 from mean Find the 90 confidence interval of the true proportion of men who applied to the nursing program Example 3 A survey of 200 000 boat owners found that 12 of the pleasure boats were named Serenity Find the 95 confidence interval of the true population of boats named Serenity Formula Sample size for Proportion z n p q 2 E 2 Example 4 A researcher wishes to estimate with 95 confidence the proportion of people who own a home computer A previous study shows that 40 of those interviewed had a computer at home The researcher wishes to be accurate within 2 of the true proportion Find the minimum sample size necessary Example 5 A researcher wishes to estimate the proportion of executives who own a car phone She wants to be 90 confident and be accurate within 5 of the true proportion Find the minimum sample size necessary 7 4 Confidence Intervals for Variances and Standard Deviations Chi Square Distribution Example 1 Find the values for when n 25 2 2 right and left for a 90 confidence interval Formula Variance n 1 s 2 n 1 s 2 2 s 2 2 c right c left d f n 1 Formula Standard Deviation n 1 s 2 n 1 s 2 s 2 2 c right c left d f n 1 Example 2 Find the 95 confidence interval for the variance and standard deviation of the nicotine content of cigarettes manufactured if a sample of 20 cigarettes has a standard deviation of 1 6 milligrams Example 3 Find the 95 confidence interval for the variance and standard deviation for the price in dollars of an adult single day ski lift ticket The data represent a selected sample of nationwide ski resorts Assume the variable in normally distributed 59 54 53 52 51 39 49 46 49 48