Transcript Estimating Confidence Intervals

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”
The z-table is used to set the lower
and upper confidence limits
-1.96
z
+1.96
• 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
100 percent of scores
95 percent of scores
2½ percent
2½ percent
Class exercise
Case #
Score
1
3
2
2
3
5
4
2
5
3
6
3
7
2
8
4
9
5
10
3
Mean
Diff.
Squared
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 parameter
somewhere 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
z scores
-1.96 -1.0
0
Sample 1
2.25
2.9
3.55
Sample 2
1.77
2.4
3.03
2½ pct.
•
•
•
•
+1.0 +1.96
95 percent of scores
2½ pct.
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.
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
n = 100
New sum of squares = 26.7
New sum of squares = 89
s2 (variance) = .92
s2 (variance) = .9
s (standard deviation) = .96
s (standard deviation) = .95
Sx (standard error of the mean) = .18
Sx (standard error of the mean) = .1
z (Sx) = .35
z (Sx) = .2
Confidence interval = 2.55  3.25
Confidence interval = 2.7  3.1
(Old Ci was 2.25  3.55)
(Old Ci was 2.25  3.55)
Final exam preview
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