Question

A survey was conducted of newlyweds in a country who have a spouse of a different race or ethnicity from their own. The survey included random samples of 1000 newlyweds in Ethnicity A and 1000 newlyweds in Ethnicity B. In the survey, 10% of respondents in Ethnicity A and 21% of respondents in Ethnicity B had a spouse of a different race or ethnicity from their own. At a=0.01, is there evidence to support the claim that the proportion of newlyweds in Ethnicity A who have a spouse of a different race or ethnicity from their own is less than the proportion of newlyweds in Ethnicity B that have a spouse of a different race or ethnicity from their own? newlyweds in Ethnicity B that have a spouso of a different race or ethnicity from their own. Stato the null and alternative hypotheses O A. Ho: P1 P2 HP2 P2 OD. Ho: P45P2 H: PP2 Calculate the standardized test statistic, OB. Ho: Pr>P2 HP SP2 O E. Ho: PP2 H:P"P2 OC. Ho: P = P2 H₂ : Pq PP2 OF. Ho: P2P2 H₂ : P1 <P2 (Round to two decimal places as needed.) Calculate the P-value (Round to three decimal places as needed.) State the conclusion of the hypothesis test. Since P a. Ho. There evidence to support the claim that the proportion of newlyweds in who have a spouse of a different race or ethnicity from their own is less than the proportion of newlyweds in Mthat have a spouse of a different race or ethnicity from their own.

Answer 1

Ans :

i)

F)

$H_{0}: p_{1} \geq p_{2}$

ii)

z = -6.80

iii)

p-value = 0.000

iv)

Since p-value < (less than ) $\alpha$ Reject H0There is sufficient evidence to support the claim that the proportion of newlyweds in Ethnicity A who have a spouse of different race or ethnicity from their own is less than the proportion of newlyweds in Ethnicity B that have a spouse of a different race or ethnicity from their own.

#######################################Explanation####################

Here we want to test whether the proportion of newlyweds in Ethnicity A who have a spouse of different race or ethnicity from their own is less than the proportion of newlyweds in Ethnicity B that have a spouse of a different race or ethnicity from their own.

Let $p_{1}$ be the proportion of respondents in Ethnicity A who had a spouse of a different race or ethnicity from their own.

Similarly, let $p_{2}$ be the proportion of respondents in Ethnicity B who had a spouse of a different race or ethnicity from their own.

The null hypothesis is given as

$H_{0}: p_{1} \geq p_{2}$

i.e the proportion of newlyweds in Ethnicity A who have a spouse of different race or ethnicity from their own is the greater ofr same as the proportion of newlyweds in Ethnicity B that have a spouse of a different race or ethnicity from their own.

i.e the proportion of newlyweds in Ethnicity A who have a spouse of different race or ethnicity from their own is less than the proportion of newlyweds in Ethnicity B that have a spouse of a different race or ethnicity from their own.

From the sample survey for newlyweds in Ethnicity A

$n_{1}=$

$\hat{p_{1}}=$

From the sample survey for newlyweds in Ethnicity B

$n_{2}=$

$\hat{p_{2}}=$

The pooled sample proportion is given as

$p=\frac{ \hat{p_{1}} * n_{1} + \hat{p_{2}} * n_{2}} { n_{1}+n_{2}}$

$=\frac{ 1000 *0.1 + 1000 * 0.21}{1000+1000}$

= 0.155

The test statistic is given as

$z =\frac{ \hat{p_{1}}- \hat{p_{2}}}{ \sqrt{ p(1-p) \frac{1}{n_{1}} + \frac{1}{n_{2}} } }$

$=\frac{0.1 - 0.21 } {\sqrt{0.155*(1-0.155)(1/ 1000 + 1/ 1000)}}$

= -6.796471

Since the hypothesis is left tailed the p-value is given as

Hence the p-value is 0.000

At the level of significance 0.01

Since the p-value < 0.01

We reject the null hypothesis.

Hence there is sufficient evidence to support the claim that the proportion of newlyweds in Ethnicity A who have a spouse of different race or ethnicity from their own is less than the proportion of newlyweds in Ethnicity B that have a spouse of a different race or ethnicity from their own.