Eleven people were given 46 grams (1.6 ounces) of dark chocolate every day for two weeks, and their vascular health was measured before and after the two weeks. Larger numbers indicate greater vascular health, and the mean increase for the participants was 1.3 with a standard deviation of 2.32. Assume a dotplot shows the data are reasonably symmetric with no extreme values. Find and interpret a 90% confidence interval for the mean increase in this measure of vascular health after two weeks of eating dark chocolate. Can we be 90% confident that the mean change for everyone would be positive? (a) What is the margin of error for the confidence interval found in Example 1?

(b) What sample size is needed if we want a margin of error within ±0.5, with 90% confidence? (Use the standard deviation from the original sample to estimate σ.)