Expert: Josh - 4/10/2009

Question
A study of iron deficiency among infants compared samples of infants following different feeding regimens. One group contained breast-fed infants, while the children in other group were fed a standard baby formula without any iron supplements. Here are summary results
On blood hemoglobin levels at 12 months of age.
Group            n      Mean     s
Breast-fed       23     13.3     1.7
Formula          19     12.4     1.8
a. Is there significant evidence that the mean hemoglobin level is higher among breastfed
babies? State null hypothesis and alternate hypothesis and conduct a t-test.
b. Give a 95% confidence interval for the mean difference in hemoglobin level between
the two populations of infants.


Answer
Hello,

You need to compute the test statistic t=(mean(x)-mu_0)/(s/sqrt(n)).
Here, set mu_0=12.4 (the mean for group using formula).

i) Null hypothesis, Ho: mu=mu_0
  Alternative hypothesis, H_alpha: mu>mu_0
ii)set significance level at 0.05, for n=23, using the t-table, the critical value t(alpha,n-1)=t(.05,22)=1.717.
iii) t=(13.3-12.4)/(1.7/sqrt(23))=2.539. Since this t-value exceeds t(alpha,n-1)=1.717, we reject the null hypothesis Ho.

Evidence supports conjecture "that the mean hemoglobin level is higher among breastfed babies"

b) Proceed on the assumption that the two population variances are equal. Then, the variable T= [mean(X)-mean(Y)-(mu1-mu2)]/[S*sqrt(1/m+1/n)] follows a t distribution with m+n-2 degree of freedom. S is the pooled estimator of the common variance sigma^2.
So, P(-t(alpha/2,m+n-2)<=T<=t(alpha/2,m+n-2))=1-alpha.

The pooled (1-alpha)*100% t-confidence interval for mu1-mu2 is
mean(x)-mean(y)+t(alpha/2,m+n-2)*s*sqrt(1/m+1/n).   [**]

one-by-one, let's work this out:
m=23, n=19,
s^2=[22*(1.7)^2+18*(1.6)^2]/(23+19-2), you need the sqrt of this.
mean(x)=13.3, mean(y)=12.4
use a table to find t(alpha/2,m+n-2) and you are done.
plug values into [**]

I hope this helps...I'm running out of time, so I won't be able to answer questions of this complexity for a while. A standard text like J.L.Devore, "Probability and statistics for engineers and the sciences" is a good reference to have. It think it would answer many of your questions.