# Other assumptions required for this procedure to be

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Other assumptions required for this procedure to be valid are:For both the confidence interval and hypothesis testing procedures, we are interested in thedifferencebetween population means:μ1μ2.IfX¯1X¯2is a statistic, then it must have a sampling distribution. If the assumptions above are met, thenthe sampling distribution forX¯1X¯2is:10.1.1Confidence Intervals for the Difference in Population MeansUsing the information about the sampling distribution from above, a (1 −α)100% confidence intervalfor the difference in population means will be of the form:Our interpretation of this confidence interval is the same as what we have done for the one populationexamples:10.1.2Hypothesis Tests for the Difference in Population MeansOur null hypothesis takes the general formH0:μ1μ2=D0, whereD0is some hypothesized differencebetween the population means.Often, we are testing the null hypothesis ofno difference, orno effect.
84There are three options for the alternative hypothesis:Ha:μ16=μ2(two-sided test)Ha:μ1< μ2(one-sided, lower/left-tail test)Ha:μ1> μ2(one-sided, upper/right-tail test) The test statistic is calculated as:IfH0is true and the assumptions are satisfied,Zwill follow a standard normal distribution.Graphically:EXAMPLE: A study was conducted to compare the starting salaries of university graduates from anhonours Bachelor’s program, versus those graduates in a general Bachelor’s program. Previous studieshad found that the population standard deviation of starting salaries for honours and general graduateswere \$2250 and \$2380 respectively, while independent random samples of size 50 from eachpopulation resulted in sample mean starting salaries of \$45,554 and \$43,348 respectively. It isreasonable to assume that the starting salaries of both honours and general degree graduates arenormally distributed.a)Does this data provide sufficient evidence, at the 5% level of significance, to suggest thathonoursgraduates make higher mean starting salaries than general graduates?
85b)What is a 95% confidence interval for the true difference in mean starting salaries between thetwogroups of graduates?10.2Two-sample procedures whenσ1andσ2are UNKNOWNUsually we do not knowσ1andσ2, in which case we have two options for inference procedures on twopopulation means:Welch’s approximatetprocedure: we don’t knowσ1andσ2, anddo notassume they are equal.Pooled variancetprocedure: even though we don’t knowσ1andσ2, weassumethey are equal.10.2.1Welch’s ApproximatetprocedureThis method requires two assumptions in order to be valid:In Welch’s procedure, we simply substitute the sample variances,and, in for the values ofand:The (1 −α)100% confidence interval forμ1μ2is then given by:In the hypothesis testing procedure, our test statistic becomes:
86If the null hypothesis is true, then the test statistic follows anapproximatetdistribution.

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