Question

please answer (a) and (b) and if possible please use R Studio when solving

Question 2 (4 points) In a certain Scandinavian population, it is suspected that 66% of individuals have blonde hair. You suspect that this percentage is actually higher. You take a random sample of 18 individuals from this population. Let X repre- sent the number of individuals in your sample who have blonde hair. Suppose you decide on a rejection region of {X > 15). (a) [2 points) Compute the probability of making a Type I error and interperet what a Type I error means in the context of this data. (b) [2 points) Suppose the actual percentage of individuals with blonde hair in this population is 72%. Compute the probability of making a Type II error and interperet what a Type II error means in the context of this data.

Answer 1

(a) Here we are to test

$H_{0}:p=0.66\;\;against\;\;H_{1}:p>0.66$

where p is the proportion of the persons in the sample with blonde hair. It is evident

X ~ Bin(18, 0.66) under H

Now the probability of type I error is given by

$P(X=15|H_{0})$

We calculate this probability using R.

R CODE:

comb = function(n, x)
{
factorial(n) / factorial(n-x) / factorial(x)
}
comb(18,15)*(0.66^15)*((1-0.66)^(18-15))

R OUTPUT:

0.06299208

Hence the probability of type I error is 0.06299208

In this context, type I error is the error commited in rejecting the hypothesis that the proportion of people with blonde hair in the given population is 0.66 even when it is true and the probability of doing this is 0.06299208.

(b)

Given the actual percentage of population with blonde hair is 72%.

(a) Here we are to test

$H_{0}:p=0.66\;\;against\;\;H_{1}:p=0.72$

where p is the proportion of the persons in the sample with blonde hair. It is evident

X ~ Bin(18, 0.66) under H

Now the probability of type II error is given by

$1-P(X=15|H_{1})$

We calculate this probability using R.

R CODE:

comb = function(n, x)
{
factorial(n) / factorial(n-x) / factorial(x)
}
1-(comb(18,15)*(0.72^15)*((1-0.72)^(18-15)))

R OUTPUT:

0.8702368

In this context, type II error is the error commited in accepting the hypothesis that the proportion of people with blonde hair in the given population is 0.66 even when it is false and the probability of doing this is 0.8702368.

Hopefully this will help you. If you are satisfied with the answer, give it a like. In case of any query, do comment. Thanks.