1. Confidence Intervals for Population Means BUSA 2100, Sections 8.1, 8.2

2. Estimation Process l Point estimates are single numbers, e.g. X-bar = $35,000. l Point estimates should be close to the true population mean (or proportion), but are almost never exactly equal. l So we will use interval estimates, known as confidence intervals. l Confidence intervals are a range of numbers, e.g. $30,000 to $40,000.

3. Confidence Intervals l Two conditions for confidence intervals: l 1. We want them to be narrow. l For example, the interval $15,000 to $55,000 is not a useful estimate for the mean salary of a firm’s employees. (too wide). l But the interval $34,000 - $36,000 is useful. l 2. We want them to have a high probability (usually >= 90%) of containing the population mean (or proportion).

4. Confidence Levels l The probability that a confidence inter- val contains the population mean or proportion is the confidence level. l The most common confidence levels are 90%, 95%, and 99%. l In a normal distribution, how many std. deviations away from the mean do we need to go to include 95% of the items?

5. Confidence Levels, Page 2 l.l.

6. Confidence Levels, Page 3 l.l.

7. Confidence Interval Formula for Means l,l,

8. Accountant Incomes Example l Example: A random sample of 64 accountants in Georgia is selected and their annual incomes are recorded. l The sample mean is $55,000 and the sample std. deviation is $4,000. l Find a 90% confidence interval for the population mean annual income.

9. Incomes Example, Page 2 l.l.

10. Incomes Example, Page 3 l.l.

11. Incomes Example, Page 4 l Part b : Find a 95% confidence interval.

12. Incomes Example, Page 5 l Part c: Find a 99% confidence interval if the sample size is 400.

13. Confidence Intervals for Small Samples or Sigma Unknown l For sample sizes < 100 or when sigma is unknown, we use the t-distribution instead of the normal curve table. (In some situa- tions, the normal curve table can still be used for sample sizes much less than 100.) l Same process, except use t-values instead of z-values. l t-values depend on the sample size, or degrees of freedom.

14. The t-Distribution l The number of degrees of freedom is 1 less than the sample size; df = n - 1. l The t-table is based upon areas in 1 tail.

15. Sample Size and t-values l Look up t-values for various confidence levels and sample sizes. l For small n and df values, the t-values are very large. This creates wide confi- ence intervals with poor accuracy.

16. Sample Size and t-values, Page 2 l.l.

17. Confidence Interval Formula for Means, Small Sample Size l Ex.: For a sample of 15 trainees, the sample mean training time is 26 hours & the sample std. deviation is 4.2 hours. l Find a 99% C.I. for the population mean training time.

18. Training Time Example l.l.