John is interested in the proportion of cars with a Japanese brand in the Califo

Question

John is interested in the proportion of cars with a Japanese brand in the California population. He tentatively believes that the proportion of Japanese cars is 0.3, but he is willing to change his belief depending on the evidence he acquires.

To judge whether he should abandon his initial belief, he is going to collect a random sample of 300 car owners.

(a) Set up the hypotheses John is going to test.

(b) John is going to reject the null hypothesis if the sample proportion exceeds a particular value PL . In determining the value of PL , he sets the probability of Type I error at 5%. What is the value of PL ?

(c) Draw the probability distribution of p under the assumption that the null hypothesis is correct. Designate the point PL

(d) Suppose that he finally collected a random sample of 300 car owners. It turns out that 105 out of 300 own cars with a Japanese brand. Would John reject the null hypothesis and conclude that the proportion of Japanese brands exceeds 0.3?

Explanation / Answer

a) hypothesis: Proportion of Japanese cars is not equal to 0.3

b) Transforming ths above statement into numerical form,

given, p =0.3, N=300

P[X>PL]=0.05

Using standardization,

P[Z> PL-p/sqrt(pq)]=0.05

P[Z> PL-0.3/sqrt(0.3*0.7)]=0.05

P[Z> PL-0.3/0.21]=0.05

using normal tables,

[PL-0.3]/0.21= 1.64

PL= 0.64

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