Breast Milk and IQ. Considerable controversy exists over whether long-term neurodevelopment is affected by nutritional factors in early life. A. Lucas and R. Morley summarized their findings on that question for preterm babies in the publication "Breast Milk and Subsequent Intelligence Quotient in Children Born Preterm (The Lancet, Vol. 339, Issue 8788, Pp, 261-264). The researchers analyzed IQ data on children at age $7\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.-8$years. The mothers of the children in the study had chosen whether to provide their infants with breast milk within $72$hours of delivery. The researchers used the following designations. Group I: mothers declined to provide breast milk; Group ll a: mothers had chosen but were unable to provide breast milk and Group Il b; mothers had chosen and were able to provide breast milk. Here are the summary statistics on IQ.

At the $1\%$significance level, do the data provide sufficient evidence to conclude that a difference exists in mean IQ at age $7\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.-8$years for preterm children among the three groups? Note: For the degrees of freedom in this exercise:

The given data is

The level of significance is $\alpha =0.01$

Let us do the test hypotheses

Null hypotheses

$H_0$: There is no difference exist in mean IQ at age years for preterm children among the three groups.

Alternative hypothesis

$H_a$: There is a difference exist in mean IQ at age years for preterm children among the three groups.

The mean of the observation is

$\overline{x}=\frac{90(92.8)+17(94.8)+193(103.7)}{90+17+193}$

$=\frac{29977.7}{300}$

$=99.926$

The treatment sum of squares is

$SSTR=\sum n_i{\left({\overline{x}}_i-\overline{x}\right)}^2$

$=90(92.8-99.926{)}^2+17(92.8-99.926{)}^2+193(92.8-99.926{)}^2$

$=7765.792$

The error sum of squares is

$SSE=\sum \left(n_i-1\right){\left(s_i\right)}^2$

$=(90-1)(15.2{)}^2+(17-1\left)\right(19.0{)}^2+(193-1)(15.3{)}^2$

$=71283.840$

Then the sum of squares is

$SST=SSTR+SSE$

$=7765.792+71283.84$

$=79049.63$

The mean treatment of the sum of squares is

$MSTR=\frac{SSTR}{k-1}$

localid="1652201913362" $=\frac{7765.792}{2}$

$=3882.896$

Then, the mean error of the sum of squares is

$MSE=\frac{SSE}{n-k}$

$=\frac{71283.840}{273}$

$=240.013$

The F-static is

$F-static=\frac{MSTR}{MSE}$

$=\frac{3882.896}{240.013}$

$=16.178$

Then, $\alpha =0.01$

Critical value is $4.68$

$F-$static $(16.178)>$critical value $(4.68)$

As a result, the crucial value approach

At a $1\%$significant threshold, the null hypothesis is rejected.

As a result, the results are statistically significant at the $1\%$level.

As a result, the results show that there is a difference in mean IQ at age years for preterm children in each of the three groups.