There has been a lot of talk in recent years about the incre

There has been a lot of talk in recent years about the increased incidence of diabetes in the US population. There have been a number of arguments that it is tied to our diet of fast and processed food and our reduced lack of activity. So the Center for Disease Control collects data on the number of emergency department visits with Hypoglycemia (high blood sugar). One of the measures that they look at is the difference in visits by gender. The following data was collected: Males Females Average number of visits 141.46 155.3 Standard deviation 3.16 4.05 Sample size 24 20

a. At the 0.02 significance level, is there a difference in the variability of the admissions between males and females?  Show work

b. Given the result from part (a) on variances, test whether there is a difference in the average admissions between genders.Use a 0.01 significance level. (If you need it, the ugly degrees of freedom are 36.) Show work

Solution

(a) Let o1^2 be the variance for males

Let o2^2 be the varaince for females

The test hypothesis:

Ho: o1^2=o2^2 (i.e. null hypothesis)

Ha: o1^1 not equal to o2^2 (i.e. alternative hypothesis)

The test statistic is

F=s1^2/s2^2

=3.16^2/4.05^2

=0.61

The degree of freedom=df1 =n1-1=24-1=23

The degree of freedom =df2=n2-1=20-1=19

It is a two-tailed test.

Given a=0.02, the critical values are F(0.01, df1=23, df2=19) =0.36 (from F table)

F(0.99, df1=23, df2=19)=2.94(from F table)

The rejection regions are if F<0.36 or F>2.94, we reject Ho.

Since F=0.61 is between 0.36 and 2.94, we do not reject HO.

So we can not conclude that there is a difference in the variability of the admissions between males and females

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(b) Let mu1 be the mean for males

Let mu2 be the mean for females

Ho: mu1=mu2

Ha: mu1 not equal to mu2

The test statistic is

t=(xbar1-xbar2)/sqrt(s1^2/n1+s2^2/n2)

=(141.46-155.3)/sqrt(3.16^2/24 +4.05^2/20)

=-12.45

It is a two-tailed test.

Given a=0.01, the critical values are t(0.005, df=36) =-2.719 or 2.719 (from student t table)

The rejection regions are if t<-2.719 or t>2.719, we reject Ho.

Since t=-12.45 is less than -2.719, we reject Ho.

So we can conclude that there is a difference in the average admissions between genders.