CHAPTER 7 Confidence Intervals and Sample Size SampleStat 200 – Elementary StatisticsIntroductionSection 7.1 Confidence Intervals for Mean ( Known or n  30) and Sample SizeSection 7.2 Confidence Intervals for the Mean When  is UnknownWhen to Use the z or t DistributionStat 200 – Elementary Statistics IntroductionEstimationSection 7.1 Confidence Intervals for Mean ( Known or n  30) and Sample SizePoint EstimateInterval EstimateThree Properties of a good estimator1.2.3.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 Student NotesCHAPTER 7 Confidence Intervals and Sample SizeSample EFormulaMaximum Error of EstimateExample 1A 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 2A 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, findthe best point estimate of the population mean and the 99% confidence interval of the population mean.Example 3A 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 µ.nzXnzX2/2/z2196 .z22 58 .Example 4The following data represent a sample of the assets(in millions of dollars) of 30 credit unions. Find the 90% confidence interval of themean.12.23 2.89 13.19 73.25 11.59 8.74 7.92 40.22 5.012.27 16.56 1.24 9.16 1.91 6.69 3.17 4.78 2.421.47 12.77 4.39 2.17 1.42 14.64 1.06 18.13 16.8521.58 12.24 2.76Sample SizeExample 5The college president wants to estimate the average age of the students at theircollege. 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, thestandard deviation of the ages is known to be 3 years.Example 6Assume 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 UnknownCharacteristics of the t-distributionSimilar to the standard normal distribution1. 2.3.4.Differs from the standard normal distribution1.2.3.Degree of FreedomFormula Example 1Find the ta/2 value for a 95% confidence interval when the sample size is 22.Example 2nstXnstX2/2/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 3The 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 Distribution7.3 Confidence Intervals and Sample Size for ProportionsProportion NotationExample 1In a recent survey of 150 households, 54 had central air conditioning. Find pˆ and qˆ , where pˆ is the proportion of households that have central air conditioning.FormulaExample 2A 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. and or 1ˆ ˆ ˆX n Xp q pn n  nqpzppnqpzpaaˆˆˆˆˆˆ2/2/Example 3A 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 ProportionExample 4A researcher wishes to estimate, with 95% confidence, the proportion of peoplewho 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 5A 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.22/ˆˆEzqpn7.4 Confidence Intervals for Variances and Standard DeviationsChi-Square DistributionExample 1Find the values for 22leftrightand for a 90% confidence interval when n = 25.Formula: VarianceFormula: Standard DeviationExample 2Find the 95% confidence interval for the variance and standard deviation of the nicotine content of cigarettes manufactured if a sample of 20 cigarettes hasa standard deviation of 1.6 milligrams.Example 3Find 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 nationwideski resorts. Assume the variable in normally distributed. 59 54 53 52 51 39 49 46 49 482 222 2( 1) ( 1). . 1right leftn s n sd f nsc c- -< <= -2 22 2( 1) ( 1). . 1right leftn s n sd f nsc c- -< <=