Question

One urban affairs sociologist claims that the proportion, p, of adult residents of a particular city who have been victimized by a criminal is at least 55%. A random sample of 245 adult residents of this city were questioned, and it was found that 119 of them had been victimized by a criminal. Based on these data, can we reject the sociologist's claim at the level of 0.05 significance? Perform a one-tailed test. Then fill in the table below.

Answer 1

The urban affairs sociologist claims that the proportion, p, of adult residents of a particular city who have been victimized by a criminal is at least 55%. So, based on the claim the hypotheses are:

Based on the hypothesis it will be a left tailed test.

Assumptions for Z-test:

1) The sample has been selected randomly.-- Satisfied.

2) n*p*(1-p)>=10, 245*0.55*(1-0.55) = 60.6375, thus satisfied

Since both the conditions are satisfied hence we use normal distribution assumptions and use Z statistic for hypothesis testing.

Given that a random sample of n = 245 adult residents of this city was questioned, and it was found that X=119 of them had been victimized by a criminal. so, the sample proportion is computed as:

Rejection region:

Based on the type of hypothesis and the significance level 0.05 the critical value for the rejection region is calculated using excel formula for normal distribution which is =NORM.S.INV(0.05), thus the Zc is computed as -1.645.

Now reject the Ho if Z<Zc.

Test Statistic:

P-value:

The P-value is computed using the excel formula for normal distribution which is =NORM.S.DIST(-2.02, TRUE), thus P-value is computed as 0.0217.

Conclusion:

Since P-value is less than 0.05 and Z <Zc hence we can reject the null hypothesis and hence we conclude that there is sufficient evidence to warrant the claim.