Inference and Confidence Intervals

Outline

• Inferring a population mean: Constructing confidence intervals

• Examining the difference between two means for the same person

• Confidence intervals for the difference between groups• Confidence intervals for medians• Summary

Constructing Confidence Intervals

• Population μ (pronounced mu) • Estimate with sample mean ( ), the plug-in principle• But with sampling error. Estimating the region which will

usually include the population mean.

• t0.05 is usually about 2, but you need to look it up the t table.• Need to know df. df = n - 1 for this test. Here 94 - 1 = 93.

n

sdtxCI 05.0%95

Example: Newton's (1998) Hostility Data

• The mean on arrival for the 94 prisoners was 28.3 with a standard deviation of 8.0.

• df = n-1 so 93, or about 90 for the t table

setxn

sdtxCI 05.005.0%95

29.9 to26.7or 6.13.28

)825(.99.13.2894

0.899.13.28%95

CI

6.13.2894

0.899.13.28%95 CI

t distribution with df = 93 and 2.5% in each tail

What does having 95% CI of 28.3 ± 1.6 mean?

We expect that about 95% of the time when a confidence interval is made that the population mean (μ) will be within the interval created. This allows us to be fairly confident that the confidence interval we calculate contains the population mean.

It is not that there is a 95% probability that μ is within the interval. This is a tricky concept.

Confidence intervals are a fundamental tool for the frequentist statistician. In the long run, you should be right (i.e., μ with the interval) about 95% of the time.

Plotting the precision of the estimate (confidence intervals) and the spread of the distribution (standard deviations).Both are in units of the original variable (here in years).

Years in prison

Fre

qu

en

cy

0 10 20 30

01

02

03

04

0

x sd

95% CI: x t0.05sd

n

Examining the Difference between Two Means for the Same Person

• Difference in means ± t0.05 times standard error• Standard error of difference using estimate of the standard

deviation of the difference.

• Just calculate a variable for the difference, and perform the calculations as you did before.

n

sdtdiff

n

sdtxxCI diffdiff

ii 05.005.0%95 21

Brewed Awakeningshttp://mybrewedawakening.com/

• • Table 5.1 Data from 10 participants comparing how much they like two different types of coffee. • • FRESHi INSTANTi DIFFi DIFFi - (DIFFi - ) 2• 5 3 2 1 1• 4 3 1 0 0• 6 5 1 0 0• 3 4 -1 -2 4• 4 4 0 -1 1• 5 3 2 1 1• 6 3 3 2 4• 3 3 0 -1 1• 5 3 2 1 1• 4 4 0 -1 1• Sum 45 35 10 0 14• Mean 4.5 3.5 1.0 0 1.56*• * This 1.56 is not the actual mean. It is the variance of the variable DIFF i, the sum of squares divided by the number

of cases minus one (14/9). We calculated it this way because it can be used in later calculations.• ________________________________________________________________________• •