Question

Preventing peanut allergies A recent study of peanut allergies—the LEAP trial—

explored the relationship between early exposure to peanuts and the subsequent

development of an allergy to peanuts. Infants ($4\mathrm{to}11$months old) who had shown

evidence of other kinds of allergies were randomly assigned to one of two groups. Group $1$consumed a baby-food form of peanut butter. Group $2$avoided peanut butter. At $5$years old, $10$of $307$children in the peanut-consumption group were allergic to peanuts, and $55$of $321$children in the peanut-avoidance group were allergic to peanuts.

a. Does this study provide convincing evidence of a difference at the $\alpha =0.05$

significance level in the development of peanut allergies in infants like the ones in this study who consume or avoid peanut butter?

b. Based on your conclusion in part (a), which mistake— a Type I error or a Type II error—could you have made? Explain your answer.

c. Should you generalize the result in part (a) to all infants? Why or why not?

Short answer

a. Convincing evidence is present that difference in development of peanut allergies for ones who consumed or avoid it.

b. Type I Error.

c. It cannot generate to all infants.

Step-by-step solution

Given Information

It is given that $x_1=10$

$x_2=55$

$n_1=307$

$n_2=321$

$\alpha =0.05$

To explain does this study convincing evidence of a difference at the α=0.05 significance level in the development of peanut allergies in infants like the ones in this study who consume and avoid peanut butter.

Claim is difference in population.

The hypothesis are:

Null: $H_0:p_1=p_2$

Alternate: $H_a:p_1\ne p_2$

$p_1$is proportion of infants who consumed a baby food form of peanut butter is allergic to peanuts and $p_2$is proportion of infants who avoided peanut butter is allergic to peanuts.

Conditions of randomness, normality and independence all are satisfied.

Hence, Sample proportion role="math" localid="1654749851889" ${\hat{p}}_1=\frac{x_1}{n_1}=\frac{10}{307}=0.03257$

${\hat{p}}_2=\frac{x_2}{n_2}=\frac{55}{321}=0.17134$

${\hat{p}}_p=\frac{x_1+x_2}{n_1+n_2}=\frac{10+55}{307+321}=\frac{65}{628}=0.10350$

Test statistics:

$z=\frac{{\hat{p}}_1-{\hat{p}}_2-\left(p_1-p_2\right)}{\sqrt{{\hat{p}}_p\left(1-{\hat{p}}_p\right)}\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}}=\frac{0.03257-0.17134-0}{\sqrt{0.10350(1-0.10350)}\sqrt{\frac{1}{307}+\frac{1}{321}}}$

$z\approx -5.71$

Probability: $P=P(Z<-5.71)\approx 0$

As $P<0.05\Rightarrow \text{Reject}{H}_0$

So, there is convincing evidence is present that difference in development of peanut allergies for ones who consumed or avoid it.

Type of error

Based of above calculations, type I error is when we reject null hypothesis ehen null hypothesis is false.

Here, we rejected null hypothesis, it is type I error.

Can we generalize result to all infants or not

We have studied infants showing other kind of allergies. It may be possible that infants with other allergies can have peanut allergy.

Hence, we cannot generalize the result to all infants.