Question

A package delivery service wants to compare the proportion of on-time deliveries for two of its major service areas. In City A, 358 out of a random sample of 447 deliveries were on time. A random sample of 320 deliveries in City B showed that 256 were on time.

1. Calculate the difference in the sample proportion for the delivery times in the two cities.

p̂ CityA−p̂ CityB = _____?

2. What are the correct hypotheses for conducting a hypothesis test to determine whether the proportion of deliveries that are on time in City A is different from than the proportion in City B?

A. H0:pA=pB, HA:pA≠pB
B. H0:pA=pB, HA:pA>pB
C. H0:pA=pB, HA:pA<pB

3. Calculate the pooled estimate of the sample proportion.

p̂ = ____?

4. Is the success-failure condition met for this scenario?

A. No
B. Yes

5. Calculate the test statistic for this hypothesis test.

  z  = ____?

6. Calculate the p-value for this hypothesis test.

p-value = ____?

7. What is your conclusion using α=0.01?

A. Do not reject H0

B. Reject H0

8. Compute a 99% confidence interval for the difference p̂ CityA−p̂ CityB

(___ , ___)

Answer 1

For sample 1, we have that the sample size is N1​=447, the number of favorable cases is X1​=358, so then the sample proportion is

For sample 2, we have that the sample size is N2​=320, the number of favorable cases is X2​=256, so then the sample proportion is

Difference in propotion

Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

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

The value of the pooled proportion is computed as

Also, the given significance level is α=0.01.

Yes, success-failure condition is met

Rejection Region

Based on the information provided, the significance level is α=0.01, and the critical value for a two-tailed test is zc​=2.58.

The rejection region for this two-tailed test is R={z:∣z∣>2.58}

Test Statistics

The z-statistic is computed as follows:

The p-value is p = 0.9756

Decision about the null hypothesis

Since it is observed that ∣z∣=0.031≤zc​=2.58, it is then concluded that the null hypothesis is not rejected.

Using the P-value approach: The p-value is p=0.9756, and since p=0.9756≥0.01, it is concluded that the null hypothesis is not rejected.

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 0.01 significance level.

Confidence Interval

The 99% confidence interval for −0.074<p1​−p2​<0.076

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