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

2. Estimating the population mean  when the population variance  2 is unknown w However is not normally distributed. w In reality, if we do not know the population mean , it is unlikely that we will know the population standard deviation . w Therefore we use the sample standard deviation, s, to estimate the population standard deviation  and hence the standard error to estimate

3. W.S Gosset showed that has a particular distribution called the student t distribution or simply the t distribution when the population from which the sample is drawn is normal. is called the t statistic

4. What if the population from which we are sampling is not normal? w The t distribution is said to be.robust. This means that the t distribution also provides an adequate approximate sampling distribution of the t statistic for moderately non- normal populations. w In actual practice, we should draw the histogram of any random variable that you are assuming is normal, to ensure that the assumption is not badly violated. w If the assumption is not satisfied at all, due to extreme skewness, we have two options: transform the data (perhaps with logarithms) to bring about a normal distribution, or use non parametric methods (studied in QBM217)

5. What do we know about the t distribution? w It looks very much like the standard normal probability density function, but with fatter tails and slightly more rounded peaks. w It is more widely dispersed than the normal probability density function. w The graph of the t probability density function changes for different sample sizes. w The t statistic has n - 1 degrees of freedom. w The similarity between the t pdf and the standard normal pdf increases rapidly, as the degrees of freedom for the t pdf increases. w The two distributions are virtually indistinguishable when the degrees of freedom exceed 30. w The values for are identical to the corresponding

6. Estimating the population mean  when the population variance  2 is unknown The (1-α)100% confidence interval for µ is given by where is the sample mean is the value of t for the given level of confidence (S&S Table 4 in appendix) is the standard deviation of the sample mean, known as the standard error

7. Example 1 – Exercise 8.14 p264 w Here we want to estimate the population mean . w The sample mean is the best estimator of . w We have sampled from a normal population therefore, will follow the t distribution.

8. Therefore the confidence interval is given by

9. Two confidence interval estimators of  We now have two different interval estimators of the population mean. The basis for determining which interval estimator to use is quite simple. If  is known the confidence interval estimator of the population mean  is If  is unknown and the population is normally distributed, the confidence interval estimator of the population mean  is When the degrees of freedom exceed 200, we approximate the required t statistic by the value.

10. Example 2 A foreman in a manufacturing plant wishes to estimate the average amount of time it takes a worker to assemble a certain device. He randomly selects 81 workers and discovers that they take an average of 29 minutes with a standard deviation of 4.5 minutes. Assuming the times are normally distributed, find a 90% confidence interval estimate for the average amount of time it takes the workers in this plant to assemble the device. What can you report to the foreman?

11. Example 1 A foreman in a manufacturing plant wishes to estimate the average amount of time it takes a worker to assemble a certain device. He randomly selects 81 workers and discovers that they take an average of 29 minutes with a standard deviation of 4.5 minutes. Assuming the times are normally distributed, find a 90% confidence interval estimate for the average amount of time it takes the workers in this plant to assemble the device. What can you report to the foreman? Since  is unknown and the population is normally distributed, the confidence interval estimator of the population mean  is

12. Therefore the confidence interval is given by We are 90% confident that the mean assembly time lies between 28.17 and 29.83 minutes.

13. Example 3 w A random sample of 26 airline passengers at the local airport showed that the mean time spent waiting in line to check in at the ticket counter was 21 minutes with a standard deviation of 5 minutes. w Construct a 99% confidence interval for the mean time spent waiting in line by all passengers at this airport. w Assume the waiting times for all passengers are normally distributed.

14. Reading for next lecture w S&S Chapter 8 Sections 8.5 - 8.7 Exercises to be completed before next lecture w S&S 8.21 8.23