Dan PiettSTAT 211-019

West Virginia University

Lecture 9

Last WeekSampling Distribution of the Sample Mean

Central Limit TheoremLarge Sample Confidence Intervals for the

population meanSmall Sample Confidence Intervals for the

population mean

Overview9.1: Confidence Intervals for a population

proportion9.2: Confidence Intervals for a difference in

population proportions9.3: Confidence Intervals for a difference in

population means

Section 9.1

Confidence Intervals on a Population Proportion

Population ProportionIn the previous class we were interested in

predicting the population mean (µ). We did this with a confidence interval.

Now we are interested in predicting p, the population proportionThe proportion of Americans that are

unemployedThe proportion of Republicans in a voting

districtHow do we do this?

Confidence Intervals on pAll confidence intervals follow the same

structurePoint Estimate ± Critical Value * Standard

ErrorThe point estimate for the population

proportion is the sample proportion ( )

x is our number of “successes” in our sampleNumber of unemployed Americans, Number of

Republicansn is our sample size

Confidence Intervals on p Cont.Point Estimate ± Critical Value * Standard Error

The Critical Value for our confidence interval will z

This is the same z values from last class (1.645, 1.96, 2.578)

This comes at a cost though

The Standard Error for our confidence interval is defined as

Putting the Pieces TogetherSo, the formula for our confidence interval on p

is…

Example:Suppose we are interested in the proportion of

WVU students who are violent with 95% confidence.

We collect a sample of 144 WVU students and discover that 12 of them are violent. Find Answer

We are 95% confident that the true proportion of WVU students who are violent is between

Section 9.2

Confidence Intervals for a difference in population proportion

Difference in Population ProportionsSuppose we are no longer interested in the true

proportion of “successes” in a population, but are instead in the difference in the proportions of two populations.The proportion of Unemployed Male Americans vs.

Unemployed Female Americans.The proportion of Republicans in Washington County

vs. Republicans in Greene County.We will refer to our sample proportions

Where n is the sample size in the sample from our first population and m is the sample size in the sample from our second

Confidence Intervals on a Difference in Population Proportions Point Estimate

Critical ValueSame z as before (1.645, 1.96, 2.578)Much like with the CI for a population

proportion:

Standard Error

Putting the pieces togetherSo the formula for the difference in population

proportions is

Example:Suppose we are interested in the difference in the

proportional of adults who get the flu with and without a vaccination. Suppose 50 adults were vaccinated and 18 got the flu. Suppose another 100 adults were not vaccinated and 48 got the flu. 90% Confidence

Find the Confidence Interval or the difference in the proportion of adults who get the flu(-.25865, .13865)

Note that 0 is in the confidence interval

Conclusions from Differences in ProportionsWith 90% confidence, we estimate that the

proportion of vaccinated adults who get the flu is between 25.9% less and 1.9% more than the proportion of unvaccinated adults who get the flu.

Important Note:Since 0 is within our confidence interval, a

difference in proportions of 0 is plausible (ie. they are the same). Effectively, the result of this experiment indicates that vaccination was not effective in reducing the chance of getting the flu.

Section 9.3

Confidence interval for the difference in means of two populations

Back to MeansWe have now looked at 3 difference confidence

intervalsPopulation MeanPopulation ProportionDifference in Population Proportions

Suppose now we are interested the difference between two separate population means rather than proportions.The difference in mean grades of males vs. females

in a statistics classThe mean travel time of one route vs. another

routeWe must assume that the standard deviations of

our populations are equal.

Case 1: Large Samples (n & m>20)Point Estimate

Critical Valuez

Standard Error

The whole Formula

Example (Large Sample)A college statistics professor wants to estimate

the difference in performance on an exam of students who had 2 or more math courses and those who had taken fewer than two match courses. The summary data is as follows:

2+: Sample of 35, mean of 84.2, stdv of 10.2<2: Sample of 45, mean of 73.1, stdv of 14.3Find a 90% CI for the difference in mean scores

(6.6, 15.6)

Case 2: Small Samples (n or m <20)Point Estimate

Critical Valuet (degrees of freedom = m + n – 2)

Standard Error

The whole Formula

Example (Small Sample)We are interested in the learning capacity of

normal field mice and against genetically modified mice. To measure the learning capacity we send mice through the maze and measuring how quickly they can make it through a second time. The following summary data was provided. We are interested in a 95% confidence interval for the difference in the means.

Normal: n = 10, mean = 20 sec, stdv = 8 secGenetic: m = 20, mean = 10 sec, stdv = 5 sec

In Summary

Important: We require that our samples are independent and random