1. In your groups, go through all four steps of a confidence interval: The “Country Taste” bread making company wants to estimate the actual weight of their 700 g bread. It is known that the government specification for the standard deviation of weight of this bread is 15 g. A random sample of 50 breads is selected, and the sample mean weight is 696 g. Construct and interpret a 90% confidence interval for the actual weight of this type of bread at “Country Taste.” Do they have reason to believe the weight of their bread is “off”?

2. Confidence Intervals for Means and the T-Distribution SECTION 8.3 DAY 2

3. So if I don’t know µ, how would I know σ? If you don’t know σ, use a t-distribution. You will use s (the sample standard deviation) in place of σ.

4. Degrees of freedom Unlike the z-statistic, the shape of the t- distribution changes based on the sample size. t(k) stands for a t-distribution with k degrees of freedom. So, if our sample size is 30, and σ is unknown, the distribution is a t-distribution with 29 degrees of freedom. This would be written t(29).

5. The t-distribution What happens to the t- distribution as n increases? How many degrees of freedom are there if n=3? If n=10? What do I mean by n=∞?

6. Characteristics of the t-distributions They are similar to the normal distribution. They are symmetric, bell-shaped, and are centered around 0. The t-distributions have more spread than a normal distribution. They have more area in the tails and less in the center than the normal distribution. That’s because using s to estimate σ introduces more variation. As the degrees of freedom increase, the t- distribution more closely resembles the normal curve. As n increases, s becomes a better estimator of σ.

7. Using Table B Now that we are studying a different distribution, we’ll use a different table. Table B gives the t-distribution critical values. Find t* for the following situations: 1.A 98% confidence interval based on n = 22 observations. 2.A 90% confidence interval from an SRS of 10 observations. 3.A 95% confidence interval from a sample of size 7.

8. Confidence Interval Formula for a T-Distribution If you do this in your calculator, you MUST name the inference procedure. In this case, one- sample t-interval.

9. Example with Confidence Intervals Here are estimates of the daily intakes of calcium (in milligrams) for 38 women between the ages of 18 and 24 years who participated in a study of women’s bone health: {808, 882, 1062, 970, 909, 802, 374, 416, 784, 997, 651, 716, 438, 1420, 948, 1050, 976, 572, 403, 626, 774, 1253, 549, 1325, 446, 465, 1269, 671, 696, 1156, 684, 1933, 748, 1203, 2433, 1255, 1100, 1425} Display the data using a histogram and then make a normal probability plot. Describe the distribution of calcium intakes for these women. Calculate the mean, the standard deviation, and the standard error. Find the 95% confidence interval for the mean. Eliminate the two largest values and recompute the 95% confidence interval. What do you notice?

10. Check your understanding P. 511

11. Robust? A confidence interval is called robust if the confidence interval doesn’t change very much when the conditions of a procedure are violated.

12. Are the t-procedures robust? In reality, very few populations are exactly normal. How does that affect our t-procedures? The t-procedures are actually quite robust against non-normality of the population IF there are no outliers.

13. What do I mean by “check the data?” Always make a plot (boxplot, histogram, normal probability plot) to check for skewness and outliers. If n is large, this step isn’t AS necessary. WHY? Because the CLT tells us that x-bar is distributed approximately normally if n is large.

14. Some practical guidelines Except in the case of small samples, the condition that the data are an SRS from the population of interest is more important than the condition that the population distribution is normal.

15. Homework P. 518 (58, 64, 68, 72, 75-78)