Question

In this chapter, we examined the difference in educational attainment between first- and second-generation Hispanic Americans and Asian Americans based on the proportion of each group with a bachelor’s degree. We present additional data from the Pew Research Center’s 2013 report, measuring the percentage of each group that owns a home (Frankfort-Nachmias, & Leon-Guerrero, 2018, p. 231).

First-Generation Hispanic Americans (N = 899), 43% own a home

Second-Generation Hispanic Americans (N = 351), 50% own a home

First-Generation Asian Americans (N = 2,684), 58% own a home

Second-Generation Asian Americans (N = 566), 51% own a home

Source: Pew Research Center, Second-Generation Americans: A Portrait of the Adult Children of Immigrants. Pew Research Center, Washington, D.C. February 7, 2013. http://www.pewsocialtrends.org/2013/02/07/second-generation-americans/

Test whether there is a significant difference in the proportion of homeowners between first- and second-generation Hispanic Americans. Set alpha at 0.05.

Test whether there is a significant difference in the proportion of homeowners between first- and second-generation Asian Americans. Set alpha at 0.01.

Reference: Frankfort-Nachmias, C., & Leon-Guerrero, A. (2018). Social statistics for a diverse society (8th ed.). Thousand Oaks, CA: SAGE Publications, Inc.

What is the difference? (between first-and second-generation Hispanic American populations)

A -2.33

B 2.33

C. 0.0198

D 0.07

Answer 1

1)

For sample 1, we have that the sample size is N_1= 899,

p^​1​ = 0.43

For sample 2, we have that the sample size is N_2 = 351

p^2 = 0.50

The value of the pooled proportion is computed as

Also, the given significance level is α=0.05.

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho: p1​=p2​

Ha:

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,

and the critical value for a two-tailed test is

z_c = 1.96

The rejection region for this two-tailed test is

R={z:∣z∣>1.96}

(3) Test Statistics

The z-statistic is computed as follows:

(4) Decision about the null hypothesis

Since it is observed that

∣z∣ = 2.33 > zc​ =1.96,

it is then concluded that the null hypothesis is rejected.

(5) Conclusion

It is concluded that the null hypothesis Ho is rejected. Therefore, there is enough evidence to claim that population proportion p1​ is different than p2​, at the 0.05 significance level.

2)

For sample 1, we have that the sample size is N_1= 2684

p^​1​=0.58

For sample 2, we have that the sample size is

N_2 = 566

p^​2​ = 0.51

The value of the pooled proportion is computed as

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: p1​=p2​

Ha:

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=0.01, 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}

(3) Test Statistics

The z-statistic is computed as follows:

(4) Decision about the null hypothesis

Since it is observed that

∣z∣=3.034>zc​=2.58,

it is then concluded that the null hypothesis is rejected.

it is concluded that the null hypothesis is rejected.

(5) Conclusion

It is concluded that the null hypothesis Ho is rejected. Therefore, there is enough evidence to claim that population proportion p1​ is different than p2​, at the 0.01 significance level.

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