Confidence Intervals

Estimating the difference due to error that we can expect between sample statistics and the population parameter

Using the error estimate to create a “confidence interval”

-1.96 z +1.96

100 percent of scores

95 percent of scores

The z-table is used to set the lower and upper confidence limits

• 95% of the area under a standard normal curve falls between -1.96z and +1.96z

• 5% of the area falls beyond +/- 1.96z

• 2½ percent of the means would fall beyond +1.96z at the right (positive) tail

• 2½ percent of the means would fall beyond -1.96z at the left (negative) tail

2½ percent 2½ percent

Case # Score Mean Diff. Squared

1 3

2 2

3 5

4 2

5 3

6 3

7 2

8 4

9 5

10 3

Class exercise

Case # Score Mean Diff. Squared

1 3 3.2 .2 .04

2 2 3.2 1.2 1.44

3 5 3.2 1.8 3.24

4 2 3.2 1.2 1.44

5 3 3.2 .2 .04

6 3 3.2 .2 .04

7 2 3.2 1.2 1.44

8 4 3.2 .8 .64

9 5 3.2 1.8 3.24

10 3 3.2 .2 .04

Sum of squares = 11.6

Variance (sum of squares/n-1) = 1.29

Standard deviation (sq. root) = 1.14

Police recruit IQ test

There is a 95% probability that the mean IQ of the population from which this sample was drawn falls between these scores:

_____________ and _____________

lower limit upper limit

Population parametersomewhere in-between

Homework assignment• Two random samples of officers tested for cynicism

• For each sample, we needed to specify the confidence interval into which the population parameter (mean of population) will fall, to a 95 percent certainty

– In social science research we don’t want to take more than five chances in 100, or 5 percent, of being wrong

• Remember that 95 percent of the cases in a normally distributed population fall between a z of -1.96 and +1.96 (meaning that 5 percent will not)

• So – always use a z of 1.96

• NOTE: Why are we doing two samples?

– FOR PRACTICE. In research we normally only draw one random sample from each group of interest.

– TO EMPHASIZE THAT THE MEANS OF RANDOM SAMPLES WILL DIFFER. The Standard Error of the Mean projects these differences to build a confidence interval into which the population mean falls.

Results

Analysis

• Our first sample had a mean of 2.9. The second sample mean was 2.4.• We used a z of 1.96, which set the probability that the population mean would fall within

our confidence interval at 95 percent• Based on sample 1, there are 95 chances in 100 that the population mean (parameter)

falls between 2.25 and 3.55. Or, there are 5 chances in 100 that it doesn’t. • Based on sample 2, there are 95 chances in 100 that the population mean (parameter)

falls between 1.77 and 3.03. Or, there are 5 chances in 100 that it doesn’t.

z scores -1.96 -1.0 0 +1.0 +1.96

95 percent of scores 2½ pct.2½ pct.

2.25 2.9 3.55

1.77 2.4 3.03

Sample 1

Sample 2

Narrowing the confidence interval• Increase the sample size!

– We tried on two sizes, 30 and 100

– This is made-up data. To keep things simple, we based the larger samples on sample 1.

– The sum of squared deviations from the mean (sum of squares, 8.9) was tripled for n =30, and multiplied by 10 for n = 100.

– The mean (2.9) was kept the same.

n = 30

New sum of squares = 26.7

s2 (variance) = .92

s (standard deviation) = .96

Sx (standard error of the mean) = .18

z (Sx) = .35

Confidence interval = 2.55 3.25

(Old Ci was 2.25 3.55)

n = 100

New sum of squares = 89

s2 (variance) = .9

s (standard deviation) = .95

Sx (standard error of the mean) = .1

z (Sx) = .2

Confidence interval = 2.7 3.1

(Old Ci was 2.25 3.55)

CONFIDENCE INTERVAL

•You will be given scores for a sample and asked to compute a 95% confidence interval into which the population mean (parameter) should fall. To do this you must compute the sample’s standard deviation and the standard error of the mean.

•You will be asked to explain in ordinary language what the confidence interval actually represents

– Here is a good answer: There are 95 chances in 100 that, based on the mean of the sample, the population mean will fall between ___ and ___.

•You will be given formulas, but know the methods by heart. Computing standard deviation is in the week 3 slide show. Standard error of the mean and confidence intervals are in the week 12 slide show.

•Remember to always use a z of 1.96 when calculating the confidence interval.

•Sample question:

– How cynical are CJ majors? We randomly sampled five and gave them an instrument to complete. On a 1-5 scale (5 is most cynical) their responses were 3, 4, 3, 4, 5. Compute and interpret the confidence interval.

– Sample mean: 3.8

– Standard error of the mean: .42

– Confidence interval: left limit 2.98, right limit 4.62

– Interpretation: 95 chances in 100 that the population mean falls between 2.98 and 4.62

Final exam preview