Chapter 8 Confidence Intervals Statistics for Business (ENV) 1

Confidence Intervals 8.1z-Based Confidence Intervals for a Population Mean: σ Known 8.2t-Based Confidence Intervals for a Population Mean: σ Unknown 8.3Sample Size Determination 8.4Confidence Intervals for a Population Proportion

Reminder: Sampling distribution – If a population is normally distributed with mean  and standard deviation σ, then the sampling distribution of is normal with mean  M =  and standard deviation – 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  = 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 4

5 In a random sampling of students of sample size=25, we are confident that 80% of the samples will have a mean between  and  , or within the interval [474.4, 525.6]

6 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 In other words:

A point estimate is a single value (sample statistic) used to estimate a population parameter.     s Point and Interval Estimates A confidence interval is a range of values in which the population parameter (say  ) is expected to be there. Usually, people consider the 95% and 99% confidence intervals.  is between ? and ?

Example 3 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. 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?

Constructing General Confidence Intervals for µ 95% CI for the (population) mean 99% CI for the (population) mean

The 95% CI for the mean is from to percent confidence interval for the population mean

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(n  30).

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) The value of t depends on the confidence level as well as the degrees of freedom (df=n-1).

There is a family of t distributions, 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. Characteristics of the t distribution It is a continuous, bell-shaped and symmetrical distribution, which is flatter than a normal distribution.

16 Distributions of the t statistic for different values of degrees of freedom are compared to a normal distribution.

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 A point estimate of the population proportion is given by the sample proportion. Confidence Interval for a Population Proportion, obtained from the sample proportion

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. EXAMPLE 4 Here, the sample proportion is p=175/500=0.35

Selecting a Sample Size Let E be the error term appear in the CI E is also known as the width of the C.I divided by 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.. Selecting a Sample Size 2

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 $ How large a sample is required?

Sample Size for Proportions 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 The formula for determining the sample size in the case of a proportion is

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.

24 “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

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.