Question

In San Francisco, a sample of 3,284 wireless routers showed that 1,182 used encryption (to prevent hackers from intercepting information). In Seattle, a sample of 1,900 wireless routers showed that 630 used encryption.

(a) Choose the appropriate hypotheses to test whether or not the population proportion of encryption is higher in San Francisco than Seattle. Assume π1 is the proportion for San Francisco and π2 for Seattle.

(b-1) Specify the decision rule at α = .01. (Round your answer to 3 decimal places.)

Reject the null hypothesis if z_calc > ???

(b-2) Find the test statistic z_calc. (Do not round the intermediate calculations and round x₁ and x₂ to the nearest whole number. Round your answer to 3 decimal places.)

Answer 1

For San francisco, we have that the sample size is n1​=3284, the number of favorable cases is x1​=1182, so then the sample proportion is p^​1​=x1/n1​​=1182/3284 ​=0.3599

For Seattle, we have that the sample size is n2​=1900, the number of favorable cases is x2​=630, so then the sample proportion is p^​2​=x2/n2​​=630​/1900 =0.3316

The value of the pooled proportion is computed as p¯​=x1+x2/n1+n2 =1182+630/3284+1900 ​=0.3495

Also, the given significance level is α=0.01.

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho:π1 ​=π2

Ha:π1 ​>π2

This corresponds to a right-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=0.01, and the critical value for a right-tailed test is zc​=2.33.

The rejection region for this right-tailed test is R={z:z>2.33}

(3) Test Statistics

The z-statistic is computed as follows:

z cal=(p^​1​−p^​2)/\sqrt(p¯​(1−p¯​)(1/n1​+1/n2​))

​​​=(0.3599−0.3316)/sqrt(0.3495⋅(1−0.3495)(1/3284+1/1900)​)

z cal ​=2.063

(4) Decision about the null hypothesis

Since it is observed that z=2.063≤zc​=2.33, it is then concluded that the null hypothesis is not rejected.

Using the P-value approach: The p-value is p=0.0196, and since p=0.0196≥0.01, 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 π1 is greater than π2​, at the 0.01 significance level.