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Statistics for Business (ENV) Chapter 8 Confidence Intervals 1 Confidence Intervals 8.1 8.2 8.3 8.4 z-Based Confidence Intervals for a Population Mean: σ Known t-Based Confidence Intervals for a Population Mean: σ Unknown Sample Size Determination Confidence Intervals for a Population Proportion Reminder: Sampling distribution – If a population is normally distributed with mean m and standard deviation σ, then the sampling distribution of is normal with mean mM = m and standard deviation M n – Use a normal curve as a model of the sampling distribution of the sample mean • Exactly, if the population is normal • Approximately, by the Central Limit Theorem for large samples Example: SAT scores • The population of scores (X) on the SAT forms a normal distribution with mean m= 500 and = 100. • In a random sample of n = 25 students, – the distribution of the sample mean is a normal distribution with a mean=500 – and standard deviation is 100 20 x n 25 4 In a random sampling of students of sample size=25, we are confident that 80% of the samples will have a mean between m - 1.28 x and m + 1.28 x , or within the interval [474.4, 525.6] 5 In other words: The probability that will be within ±25.6 of µ is 80%, OR If we know then there is 80% probability that µ will be within ±25.6 away from OR we are 80% confident that µ will be within an interval ±25.6 away from 6 Point and Interval Estimates A confidence interval is a range of values in which the population parameter (say m) is expected to be there. Usually, people consider the 95% and 99% confidence intervals. m is between ? and ? A point estimate is a single value (sample statistic) used to estimate a population parameter. m s Example 3 The Dean of the Business School wants to estimate the mean number of hours students studied per week. A sample of 49 students showed a mean of 24 hours with a sd of 4 hours. What is the population mean? The value of the population mean is unknown. Our best estimate of this value is the sample mean of 24.0 hours. This value is called a point estimate. X m P 0 1.96 47.5% / n X m P 1.96 1.96 95% / n 1.96 1.96 P X m 95% n n 1.96 1.96 P X mX 95% n n 1.96(4) 1.96(4) P 2.4 m 2.4 95% 49 49 1.96(4) 1.96(4) P 2.4 m 2.4 95% 49 49 P (22.88 m 25.12) 95% Constructing General Confidence Intervals for µ X z n 95% CI for the (population) mean X 1.96 n X 2.58 n 99% CI for the (population) mean 95 percent confidence interval for the population mean X 1.96 24.00 1.96 n 4 49 24.00 1.12 The 95% CI for the mean is from 22.88 to 25.12. What if we don’t know ? However, normally, we don’t know the population (sd) . So, normally, people just replace (estimate) the by s. It doesn't matter too much since normally we are considering a large sample(n30). Factors that determine the width of a confidence interval The sample size, n The level of confidence, 1- The s.d. of the population, (usually estimated by the sample s.d., or s) Constructing General Confidence Intervals for µ Confidence interval for the mean (n < 30 and the underlying distribution is normal) X t s n The value of t depends on the confidence level as well as the degrees of freedom (df=n-1). Characteristics of the t distribution It is a continuous, bell-shaped and symmetrical distribution, which is flatter than a normal distribution. There is a family of t distributions, determined by its degrees of freedom (n-1). The t-distribution approaches N(0, 1) as n approaches infinity. Distributions of the t statistic for different values of degrees of freedom are compared to a normal distribution. 16 Confidence Interval for a Population Proportion Let X ~ Bin(N, ), then P =X/N is called a population proportion. The distribution for a population proportion. Both n and n(1- ) > 5 (1 ) P ~ N , A point estimate of the population proportion is given by the sample proportion P . Confidence Interval for a Population Proportion, obtained from the sample proportion P N P(1 P) Pz n EXAMPLE 4 A sample of 500 executives who own their own home revealed 175 planned to sell their house after they retire. Develop a 98% CI for the proportion of executives that plan to sell their house. Here, the sample proportion is p=175/500=0.35 (.35)(.65) 0.35 2.326 .35 .0497 500 Selecting a Sample Size Let E be the error term appear in the CI Ez n E is also known as the width of the C.I divided by 2. Selecting a Sample Size z n E 2 2 where n is the size of the sample E is the allowable error z the z- value corresponding to the selected level of confidence the population s.d.. Example 6 A consumer group would like to estimate the mean monthly electricity charge for a single family house in July within $5 using a 99 percent level of confidence. Based on similar studies the s.d. is estimated to be $20.00. How large a sample is required? 2 (2.58)( 20) n 107 5 Sample Size for Proportions The formula for determining the sample size in the case of a proportion is Z n p(1 p) E where p is the estimated proportion, based on past experience or a pilot survey z is the z value associated with the degree of confidence selected E is the maximum allowable error the researcher will tolerate 2 Example 7 The American Kennel Club wanted to estimate the proportion of children that have a dog as a pet. If the club wanted the estimate to be within 3% of the population proportion, how many children would they need to contact? Assume a 95% level of confidence and that the club estimated that 30% of the children have a dog as a pet. 2 1.96 n (.30)(.70) 897 .03 “The nationwide telephone survey was conducted Friday through Wednesday with 1,224 adults and has a margin of sampling error of plus or minus three percentage points.” Nation’s Mood at Lowest Level in Two Years, Poll Shows By JIM RUTENBERG and MEGAN THEE-BRENAN Published: April 21, 2011 24 Chapter 8 Estimation and Confidence Intervals When you have completed this chapter, you will be able to: ONE Define what is meant by a point estimate. TWO Construct a confidence interval for the mean when the population standard deviation is known and the sample size is large enough or underlying distribution is normal. THREE Construct a confidence interval for the mean when the population standard deviation is unknown and sample size is large enough or underlying distribution is normal. Chapter 8 continued FOUR Construct a confidence interval for the population proportion. FIVE Construct a confidence interval for the mean when the population size is finite. SIX Determine the sample size for attribute and variable sampling.