Question

Comparisons of two population proportions when the hypothesized difference is zero
Carry out a two-tailed test of the equality of banks’ share of the car loan market in
1980 and 1995.
Population 1: 1980
n1 = 1000
x1 = 53
? 1 = 0.53
Population 2: 1985
n2 = 100
x2 = 43
? 2= 0.53

Answer 1

since for population 1:1980

you have written p1=0.53

I assume n1=100 and not 1000

and for population 2:1985

since the total are=100 and favourable are=43

so i assume that p2=0.43 and not 0.53

The solution is as follows:

Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho:p1​=p2​

Ha:p1≠p2

This corresponds to a two-tailed test, for which a z-test for two population proportions needs to be conducted.

(2) Rejection Region

Based on the information provided, the significance level is α=0.05\alpha = 0.05α=0.05, and the critical value for a two-tailed test is zc=1.96

The rejection region for this two-tailed test is R={z:∣z∣>1.96}

(3)Test Statistics

For sample 1, we have that the sample size is N1=100, the number of favorable cases is X1=53​, so then the sample proportion is

For sample 2, we have that the sample size is N2=100 the number of favorable cases is X2​=43, so then the sample proportion is

The value of the pooled proportion is computed as

Also, the given significance level is α=0.05.

The z-statistic is computed as follows:

(4) Decision about the null hypothesis

Since it is observed that ∣z∣=1.415≤zc​=1.96, it is then concluded that the null hypothesis is not rejected.

Using the P-value approach: The p-value is p=0.157, and since p=0.157≥0.05, it is concluded that the null hypothesis is not rejected.

(5) Conclusion

It is concluded that the null hypothesis Ho is not rejected. Therefore, there is not enough evidence to claim that the population proportion p1​ is different than p2​, at the 0.05 significance level.

please rate my answer and comment for doubts.