1. 1 Confidence Interval for a Mean

2. 2 Given A random sample of size n from a Normal population or a non Normal population where n is sufficiently large. A population at least 20 times the sample size n.

3. 3 Confidence Interval for a Mean Result A confidence interval is given by where t* is the appropriate critical value for the T distribution with (n – 1) DF.

4. 4 Example Rolls (single rolls) of paper leave a factory with weights that are Normal with unknown mean. n = 8 rolls are randomly selected 1494.51483.81512.31507.0 1503.61504.51495.41521.5

5. 5 Example Rolls (single rolls) of paper leave a factory with weights that are Normal with unknown mean. n = 8 rolls are randomly selected, their sample mean weight is 1502.83 pounds; the sample standard deviation is 11.62 pounds. Determine a 95% confidence interval for the population (or true) mean weight.

6. 6 Example 0.025 0.95 0.025 t* = 2.365 T7T7

7. 7 Example Rolls (single rolls) of paper leave a factory with weights that are Normal with unknown mean. n = 8 rolls are randomly selected, their sample mean weight is 1502.83 pounds; the sample standard deviation is 11.62 pounds. Determine a 95% confidence interval for the population (or true) mean weight.

8. 8 Interpretation 1502.83  9.72 1502.83 – 9.72 = 1493.11 1502.83 + 9.72 = 1512.55 (1493.11, 1512.55) 95% of all samples* produce an interval that covers the true mean . We have the interval from one sample, chosen randomly. We are 95% confident that the mean weight of all rolls is between 1493.11 and 1512.55 pounds. * Provided sampling is done from a Normal distribution. Because the sample is small, if this requirement is not met, the confidence is not really 95%.

9. 9 TRUE / FALSE QUIZ 1. There’s a 95% confidence a roll is between 1493.11 and 1512.55 pounds. 1494.51483.81512.31507.0 1503.61504.51495.41521.5

10. 10 TRUE / FALSE QUIZ 1. There’s a 95% confidence a roll is between 1493.11 and 1512.55 pounds. 1494.51483.81512.31507.0 1503.61504.51495.41521.5 For this sample: 75%. 95% would be impossible (although closest would be “all” – which is clearly not the case).

11. 11 TRUE / FALSE QUIZ 1. There’s a 95% confidence a roll is between 1493.11 and 1512.55 pounds. FALSE: If you want to make this kind of statement, construct a prediction interval. One way of forming a 95% prediction interval is with the interval from the 2.5 th to the 97.5 th percentiles. (For a data set with 8 observations, this is tough. There are other ways…)

12. 12 Example Rolls (single rolls) of paper leave a factory with weights that are Normal with unknown mean. n = 8 rolls are randomly selected 1494.51483.81512.31507.0 1503.61504.51495.41521.5

13. 13 TRUE / FALSE QUIZ 2. There’s a 95% confidence the sample mean is between 1493.11 and 1512.55 pounds. FALSE. The sample mean definitely is between them; we can be 100% confident in that, because the sample mean centers the interval.

14. 14 TRUE / FALSE QUIZ 3. There’s a 95% confidence another random sample of rolls will have mean between 1493.11 and 1512.55 pounds. FALSE: The confidence interval estimates the population mean weight.

15. 15 TRUE / FALSE QUIZ 4. There’s a 95% probability that the mean weight of all rolls is between 1493.11 and 1512.55 pounds. FALSE (but in a way “closest” to true). The probability is either 0 or 1 – depending on what the population mean is.

16. 16 Interpretation What “95% confidence means” 95% of all samples produce an interval that covers the true mean . We have an interval from one sample, chosen randomly. Our interval either does or does not cover  : in practice we just don’t know. We do know that the procedure works 95% of the time. Interpreting a 95% confidence interval We are 95% confident that the mean weight of all rolls is between 1493.11 and 1512.55 pounds.