Confidence Intervals. Class 23. November 29, 2011
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1 Confidence Intervals Class 23 November 29, 2011
2 Last Time When sampling from a population in which 30% of individuals share a certain characteristic, we identified the reasonably likely values for the number of people in our sample who would have that characteristic.
3 Last Time When sampling from a population in which 30% of individuals share a certain characteristic, we identified the reasonably likely values for the number of people in our sample who would have that characteristic. We did this by calculating the 5th and 95th percentile of the probability histogram of the number of people in our sample who have the characteristic.
4 Last Time When sampling from a population in which 30% of individuals share a certain characteristic, we identified the reasonably likely values for the number of people in our sample who would have that characteristic. We did this by calculating the 5th and 95th percentile of the probability histogram of the number of people in our sample who have the characteristic. We found that a sample of size 40 would have anywhere from 7 to 17 individuals with the characteristic in it.
5 Reasonably Likely Values for Number of Successes in Sample 0% 0 10% % % % % % % % % % 40
6 Plausible Values for the Population Percentage In a S.R.S. of size 40 drawn from a population, it is found that exactly 30 (or 75%) are men. Is it plausible that if we were to count the number of men in the entire population, that 50% of them would be men? What percentages are plausible?
7 What Percentages are Plausible?
8 What Percentages are Plausible?
9 What Percentages are Plausible? Any value from about 65% to 85% seems plausible.
10 Confidence Intervals The horizontal segments (in magenta) were reasonably likely intervals for the number of men in a sample of size 40.
11 Confidence Intervals The horizontal segments (in magenta) were reasonably likely intervals for the number of men in a sample of size 40. In this example, we used the middle 90% to represent what seems reasonably likely.
12 Confidence Intervals The horizontal segments (in magenta) were reasonably likely intervals for the number of men in a sample of size 40. In this example, we used the middle 90% to represent what seems reasonably likely. The vertical segment (in blue) shows us the plausible values for the proportion of men in the population.
13 Confidence Intervals The horizontal segments (in magenta) were reasonably likely intervals for the number of men in a sample of size 40. In this example, we used the middle 90% to represent what seems reasonably likely. The vertical segment (in blue) shows us the plausible values for the proportion of men in the population. The vertical segment is called a 90% confidence interval for the proportion of men in the population.
14 Confidence Intervals The horizontal segments (in magenta) were reasonably likely intervals for the number of men in a sample of size 40. In this example, we used the middle 90% to represent what seems reasonably likely. The vertical segment (in blue) shows us the plausible values for the proportion of men in the population. The vertical segment is called a 90% confidence interval for the proportion of men in the population. Activity: Handout 20, problem 1.
15 The Geometry of Confidence Intervals
16 From Concept to Calculation The horizontal and vertical segments are approximately equal to each other in length over the range successes in the sample.
17 From Concept to Calculation The horizontal and vertical segments are approximately equal to each other in length over the range successes in the sample. This is because the end points of the horizontal segments lie on two parallel lines of slope 1 over this range.
18 From Concept to Calculation The horizontal and vertical segments are approximately equal to each other in length over the range successes in the sample. This is because the end points of the horizontal segments lie on two parallel lines of slope 1 over this range. A general rule of thumb is that as long as the observed number of successes and failures are at least 10, the vertical and horizontal segments will have the same end points.
19 From Concept to Calculation The horizontal and vertical segments are approximately equal to each other in length over the range successes in the sample. This is because the end points of the horizontal segments lie on two parallel lines of slope 1 over this range. A general rule of thumb is that as long as the observed number of successes and failures are at least 10, the vertical and horizontal segments will have the same end points. Activity: Handout 20, problem 2.
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