Question

The Food and Drug Administration (FDA) is asked to approve a new drug. The new drug should contain less than 25mg of the active ingredient “toxin”, which is assumed to have dangerous side effects. The FDA would like to restrict the error of “approving the drug despite its too high content of toxin” to a maximum risk of 5% (α ≤ 0.05). Let Xi : “ The content of toxin in the i-th pill [in mg].” ∼ N(µ, σ2 ) ∼ N(µ, 4). A simple random sample of n = 50 pills ( Xi ∼ i.i.d.) will be used for the test.

7. Given a significance level of α = 5% what is the highest probability of making a type II error?

8. In the sample, x¯ = 24.6. Compute the p-value. What do you conclude? [Write down the probability that you computed.]

9. Has a type I error occurred? Explain your answer.

10. Has a type II error occurred? Explain your answer.

Answer 1

7. Given , is the probability of type 1 error that is error of rejecting a true null hypothesis

Type 2 error is the error of accepting a false null hypothesis

The highest probability of type 2 error = 1- 0.05 = 0.95

Highest probability of type 2 error is 95%

8.The null and alternative hypotheses

Test statistic

= -1.41

P value = P( z < -1.41)

= 0.0787

Since P value > 0.05

We fail to reject H0.

There is not sufficient evidence to conclude that new drug contains less than 25 mg of toxins .

Note : we get the value from z table for area from to -1.41

Or excel formula " =NORM.S.DIST(-1.41,TRUE)"

9. Type 1 error is the error of rejecting a true null hypothesis

Since we fail to reject H0 , type 1 error has not occurred .

10. Since P value > 0.05

We fail to reject H0.

That is we accept H0.

Now if H0 is true , we are correct in our decision

If H0 is false , we commit error

Accepting a null hypothesis , if it is false is type 2 error

So type 2 error has occurred.