Unit 222 Measuring The Social World

SOC 222 -- MEASURING the SOCIAL WORLD
Session #6 -- CONFIDENCE INTERVALS
Oct 2013

TODAY’S OBJECTIVES

1. Understand what confidence intervals tell us
2. Understand how we get them
3. Know the SPSS procedure for getting confidence intervals

Terms to Know

Confidence interval
Mu (μ) t-curve normal curve

RICK’S SITUATION

Recap

His RQ:

Population:
Real population mean:

Sample:
Sample mean: 6

Sampling distribution: 2

Estimated mean = sample mean = 6

Estimated standard error =

CONFIDENCE INTERVALS

Way for rick to talk about how accurate he is
Accuracy in inferential stats

Confidence Interval Statements

Marginal error statements
Election polls

“The Green Party has 12 % of the vote, plus or minus 3%, 19 times out of 20”

EG #1
“The average mark for fourth-year students at UofT is 76.4%, plus or minus 4.2%, 95% of the time”.

Interpretation:

Mean

Accuracy
Lower limit: mean minus margin-of-error (MOE) = 76.4% - 4.2% = 72.2%

Upper limit: mean plus margin-of-error (MOE) = 76.4% + 4.2% = 80.6%

EG #2

“The mean days absent is 6 days, plus or minus 2.5 days, 90% of the time.”

Interpretation:

Mean 76.4 %

Accuracy

Margin of error MOE Given as an interval as a range +or – 4.2%

Lower limit: mean minus MOE = 6 - 2.5 = 3.5 days
Upper limit: mean plus MOE = 6 + 2.5 = 8.5 days

This range is the confidence interval

FINDING CONFIDENCE INTERVALS
A range where you think the mean is going to lie 95% sure the mean will lie within the intervals A way to talk about the accuracy of a population estimate
Four Steps:
1. Decide on a level of confidence :95% most common
2. Estimate the sampling distribution : info from sample
3. Locate the 95% of samples closest to the mean
4. The lower and upper boundaries of this group of samples mark the confidence interval: lower limits and higher limits that’s CI

95% is the most common in research

Step #2: Estimating the Sampling Distribution

Terminology estimated population mean

μ (mu)

Sample mean is 
Population mean is μ

Sampling Distribution Standard Error

Symmetrical

Always symmetrical

Never skewed

Step #3: Locating the Closest Samples

A sample of size 3

There are 20 possible samples: eac box is a sample

3 4 5 6 7 8 9
Mean: still 6

Standard error: + or - 3

Step #4: Finding the Confidence Interval

The closest 18 range from 4 to 8

90% confidence interval: 90% of 20 give us lower boundary and upper boundary

But what if his sample had a larger standard deviation:

1 2 3 4 5 6 7 8 9 10 11

Mean: 90% ci
Closest 18 possible range from 2-10

90% confidence interval standard deviation + or - 4

The closest 18 possible samples range from 2 to 10

2 to 10 days absent, 90% of the time.

Much less accurate

Nice tight Conf. In

Sampling Distribution Shapes 167 171 172 176

.estimating a curve

For a sampling distribution of means, it’s a bell-shaped curve.

the area under the curve.

The size of the area corresponds to the number of possible samples

Here’s an example showing the curve, and some of the samples underneath it.
Calculate tables that give us the area under the curve
This curve corresponds to the number of samples

1. For small samples, a “t” curve
a. Flatter more spread out
b. More accurate confidence interval
c. Small is around 100

2. For large samples, the normal curve
a.

what proportion of all possible samples lie in certain ranges under the curve.
Standardized scores calculating and converting Effect of Sample Size

More accurate population estimates
Narrow confidence interval
Political reporting

increase sample size

As N gets larger, the fraction gets smaller

As the fraction gets smaller, SE gets smaller

As SE gets smaller, the sampling distribution gets narrower

As the sampling distribution gets narrower, more samples become closer to the mean

So the best 95% of the samples are closer to the mean

So the confidence interval for these samples is closer to the mean

So the confidence interval shrinks, and population estimates get more precise and accurate.

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SPSS: Confidence Intervals

This procedure finds statistics about a single variable from a sample, and estimates confidence intervals for the variable.

1. Menu bar, Analyze, Descriptive Statistics, Explore
Opens the “Explore” Box
Variable list on left
Three working areas in the middle
Ignore the second and third
Option buttons on the right
A “Display” area underneath
Five action buttons on the
bottom

2. Move your variable into the top working area

3. Click the “Statistics” option button
Opens the “Explore: Statistics” sub box
Select “Descriptives” and the size of the confidence interval you want
95% is the default
Click “Continue”
Takes you back to the Explore box

4. In the “Display” area, select “Statistics”

5. OK
Opens your output file
Two tables

First table: gives you your sample size

Second table:

The mean of your sample
Also the standard error of the mean

The lower and upper bounds for a 95% confidence interval for that mean

The median

The variance

The standard deviation

The range

The interquartile range

Ignore all the “Bootstrap” results

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SUMMARY of TODAY’S OBJECTIVES

1. 3 measures of central tendency: mode, median, mean
2. 3 measures of variation: percent non-modal, range, variance
3. Getting central tendency and variation in SPSS
4. Distributions, skewness, and outliers

Proportions re category variables
Proportion of hom. By firearm and other
Etimats of pop proportion
2 categories proportions are decimals

dummy variables special kind of dichotomy coded as 1 and 0 use 1 for the cat of interest
0 for the other

advantage spss mean of dummy var. gives you proportion confidence interval proportion code differently in values
.93
.98 proportions for cat variable