QBM117 Business Statistics

1. QBM117 Business Statistics Estimating the population mean , when the population variance 2, is known

2. Statistical inference • Statistical inference is the process by which we acquire information about populations from samples. • We cannot be certain our conclusions are correct - a different sample might lead to a different conclusion. • Statistical inference uses probability, to indicate how trustworthy its conclusions are. • There are two procedures for making inferences: • interval estimation. • hypothesis testing.

3. In the lectures this week, we will learn how to calculate intervals to estimate the unknown population parameters. The approach we will use will be • to identify the parameter to be estimated. • to specify the parameter’s estimator and its sampling distribution. • to construct an interval estimator.

4. We will develop techniques to estimate two population parameters. • The population mean m • The population proportion p (for qualitative data) • Examples • A bank conducts a survey to estimate the mean number of times customers actually use ATM machines. • A random sample of processing times is taken to estimate the mean production time on a production line. • A random sample of customers are surveyed to estimate the true proportion of customers who use EFTPOS to pay for their purchases.

5. Concepts of Estimation • The objective of estimation is to determine the value of a population parameter on the basis of a sample statistic. • There are two types of estimators • a point estimator • an interval estimator

6. Point Estimator • A point estimator draws inference about a population by estimating the value of an unknown population parameter using a single value or a point. • For example, the sample mean is a point estimator of the population mean • The sample proportion is a a point estimator of the population proportion. Interval Estimator • An interval estimator draws inferences about a population by estimating the value of an unknown population parameter using an interval. • The interval estimator is affected by the sample size.

7. Selecting the right sample statistic to estimate a population parameter value depends on the characteristics of the statistics. • Estimator’s desirable characteristics • Unbiasedness: An unbiased estimator is one whose expected value is equal to the parameter it estimates. • Consistency: An unbiased estimator is said to be consistent if the difference between the estimator and the parameter grows smaller as the sample size increases. • Relative efficiency: For two unbiased estimators, the one with a smaller variance is said to be relatively efficient.

8. Interval estimation • In interval estimation, an interval is constructed around the point estimate, and we state that this interval is likely to contain the true population parameter. • To this point in time, we have been using the sample mean as our best estimate of the population mean. With interval estimation, we obtain an interval by adding a number to, and subtracting the same number from the sample mean. We then state that the interval is likely to contain the population mean. This procedure is called interval estimation.

9. So how do we know what number to add to or subtract from our point estimate? • This depends on two things: • the standard deviation of the sample mean; • the level of confidence to be attached to the interval. The larger the standard deviation of the mean, the larger the number added to or subtracted from, the mean. The higher the level of confidence we desire from our interval, the larger the number added to or subtracted from the mean.

10. Confidence level and confidence interval • Each interval is constructed with a certain level of confidence. • The confidence level associated with a confidence interval states how much confidence we have that this interval contains the true population parameter. • The confidence level is denoted (1 - )100% where (1 - ) is the probability the interval will contain  and  is the probability that it will not. • Although any confidence level can be chosen, the more common values are 90%, 95% and 99%.

11. Estimating the population mean  when the population variance 2 is known The (1-α)100% confidence interval for µ is given by where is the sample mean is the value of z for the given level of confidence (S&S table 8.1) is the standard deviation of the sample mean, known as the standard error

12. Example 1 – Exercise 8.1 p255 • Here we want to estimate the population mean . • The sample mean is the best estimator of . • We have no information about the population from which we are sampling, however, a sample of size 400 (at least 30) has been selected. • Therefore, will be approximately normally distributed with a mean  and standard deviation

13. Therefore the confidence interval is given by

14. Interpreting the confidence interval • Either the interval 74.02 - 75.98 contains the true mean  or it doesn’t. • We cannot know whether our sample is one of the 95% for which the interval 74.02 - 75.98 captures the true . • We know that 95% of all such intervals will capture the true mean and 5% will not. • Therefore a 95% confidence interval will capture the unknown  in 95% of all possible samples.

15. Example 2 A public health official wanted to know how often university students visit the medical centre on campus. The standard deviation number of visits per student is believed to be 0.44. The official took a random sample of 1200 students nationwide and found an average of 2.3 visits per student per year. Find a 90% confidence interval estimate for the mean number of visits per year for all university students.

16. Reading for next lecture • S&S Chapter 8 Section 8.4 Exercises to be completed before next lecture • S&S 8.5 8.9