91影视

It is given that $x_1=10$

$x_2=55$

$n_1=307$

$n_2=321$

$伪=0.05$

Claim is difference in population.

The hypothesis are:

Null: $H_0:p_1=p_2$

Alternate: $H_a:p_1鈮�p_2$

$p_1$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:

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

$z鈮�-5.71$

Probability: $P=P(Z<-5.71)鈮�0$

As $P<0.05鈬�\text{Reject}{H}_0$

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

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.

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.