Basic Statistics Inferences About Two Population Means

STRUCTURE OF STATISTICS STATISTICS DESCRIPTIVE INFERENTIAL TABULAR GRAPHICAL NUMERICAL ESTIMATION TESTS OF HYPOTHESIS

A social psychologist wanted to determine if the development of “generosity”was related to the gender of children. As a pilot study the psychologist obtained a random sample of 4-year- old boys and girls. In a group setting, each child was given 16 small pieces of candy and asked to “put some in a sack for your very best friend.” The numbers of pieces of candy set aside for friend by 10 girls and 10 boys are shown below: Research situation for independent two-samples t-Test

The Research Design In a group setting, each child was given 16 small pieces of candy and asked to “put some in a sack for your very best friend.” The numbers of pieces of candy set aside for the friend by 10 girls and 10 boys are shown below: Gender BoyGirl n=10 X= 10 n=10 X=12 DV:Generosity

Step by Step: The Two-Sample Test of Hypothesis Using the t-test. 1.State Research Problem or Question 2.Establish the Hypotheses 3.Establish Level of Significance 4.Collect Data 5.Calculate Statistical Test 6.Interpret the Results

State the Research Hypothesis There is a difference between 4-year-old boys and girls in their levels of generosity. Boy Girl Generosity Dependent Variable Gender Independent Variable difference

The research question is “Is there is a difference in the development of generosity between boys and girls. Research or Alternative Hypothesis Therefore, the research hypothesis is:

Setting the Null Hypothesis The null hypothesis is set by “nullifying” the research hypothesis. Since μ boy = μ girl can be written as μ boy – μ girl = 0, the null hypothesis can be written:

Generosity 4-year-old BOYS GIRL boys girls n=10 Population Random sampling Measurement of DV 4-year-old GIRLS Random Sample Generosity Calculation of mean difference ? Difference Research Hypothesis

Identify the Test Statistic We will be using Confidence Intervals to test the hypotheses about the differences in two population means.

We have already seen that we can estimate the individual population means with sample means. We have also seen that the null hypothesis can be written as follows: If we consider μ 1 – μ 2 the parameter we are estimating, we can estimate it with: While we do not know the sampling distribution of the difference, we do know it for both of the sample means individually. We must find out how to combine them. We will illustrate how it might be done with a simple example.

Consider the following example: X2X2 f X1X1 f We have the following two distributions of X 1 and X 2 : We are going to combine these two distributions into one distribution of (X 1 - X 2 ): 1-2 = -1 X 1 - X 2 f = = = 1 1 Range of X 1 = 2 – 1 = 1 Range of X 2 = 2 – 1 = 1 Range of (X 1 – X 2 ) = 1 – (-1) = 2 Mean of X 1 - X 2 = 0

What do we know for the problem at hand? 1. From the CLT, we know that the sample means from the population of boys and the population of girls (sampling distributions) are distributed approximately normally. 2. We know that the means of the original distributions of boys and girls (μ boys and μ girls ) have the same population means as the sampling distributions. 3. We also know that the standard deviations of the sampling distributions are the same as those in the original distributions of boys and girls, except they are divided by the square roots of the sample sizes. 4. Finally, we know from the demonstration on the previous slide that the mean of the difference is the difference in the means and variability of the difference is the sum of the variability of the individual distributions.

0 Standard error of difference for independent-samples Deriving a Sampling Distribution of Mean Difference =

Calculating the “Pooled” Variance This variance is referred to as the “pooled” variance since it contains the appropriate (weighted by the sample sizes) amount of information from each of the two samples.

Testing with Confidence Intervals and t-Test The formula for the confidence interval for two independent samples is: The formula for the two-sample t-test is Note that  1 -  2 is hypothesized to be 0!

Conducting the Statistical Test: We will use the 95% Confidence Interval From our problem: n boys = 10 n girls = 10 = 12 = 10 S boys = 2.5 S girls = 3.0 =

The 95% Confidence Interval = (1.23) = = 4.78 and = 2 – 2.78 = We are 95% confident that the mean difference between boy’s and girl’s generosity is between –0.78 and Since 0 is in the interval, we accept the Null Hypothesis of no difference in generosity.

0 A Graphical Representation of Results Sampling distribution of mean differences % Confidence Interval

The “Dependent” Samples t-test The previous example assumed independent random sampling. What if the two samples are dependent on each other?

An Example Assume that the government plans to evaluate its campaign to conserve gasoline. Twelve families are randomly selected and their gasoline consumption is measured before and after the campaign. The data are presented on the next slide. This problem is on page 322 of your text using the t-statistic. Compare the answers!

The Data FamilyBeforeAfterDifferenceDifference 2 A B C D E35361 F G H I48491 J K L Total  d = 35  d 2 = 235

In essence, we will treat the differences in the two samples as if we were calculating a one-sample confidence interval. We must calculate the mean difference d and the standard deviation of the differences S d Confidence Interval for Dependent Samples

Calculating the 95% Confidence Interval Thus, we estimate, at a 95% level of confidence, that the real difference is between 4.12 and gallons and we reject the Null Hypothesis and conclude the campaign did affect gas consumption. (see page 324)

Summary of Two Sample Tests We can use confidence intervals to test an hypothesis about the difference in two independent samples. We can also use confidence intervals to test an hypothesis about the difference in two dependent samples. The conclusions reached using confidence intervals are exactly the same as using the t- statistic.