Question

A concerned group of citizens wanted to know if the proportion of forcible rapes in Texas was different in $2011$

than in $2010.$Their research showed that of the $113,231$violent crimes in Texas in $2011,7622$of them were forcible rapes. In $2011,$$7439$of the $104,873$violent crimes were in the forcible rape category. Test at a $5\%$significance level.

Answer the following questions:

a. Is this a test of two means or two proportions?

b. Which distribution do you use to perform the test?

c. What is the random variable?

d. What are the null and alternative hypothesis? Write the null and alternative hypothesis in symbols.

e. Is this test right-, left-, or two-tailed?

f. What is the p-value?

g. Do you reject or not reject the null hypothesis?

h. At the ___ level of significance, from the sample data, there ______ (is/is not) sufficient evidence to conclude that

____________.

Short answer

a. This is a test of two proportions.

b. The distribution to perform the test is the normal distribution.

c. The random variable is $X.$

d. $$\begin{gathered} H_0:p_1=p_2 \\ {H}_a:p_1\ne p_2 \end{gathered}$$

e. The test is two-tailed.

f. The p-value is $0.069.$

g. We do not reject the null hypothesis.

h. At the $5\%$ level of significance, from the sample data, there issufficient evidence to conclude that the proportion of forcible rapes in Texas in $2011$and $2010$are the same.

Step-by-step solution

Given information

The research showed that $113231$violent crimes in Texas in $2010,7622$of them were forcible rapes. In $2011,7439$of the $104873$violent crimes was in the forcible rape category. Test at $5\%$significant level.

Explanation (part a)

This is a test of two proportions.

Explanation (part b)

The distribution to perform the test is the normal distribution.

Explanation (part c)

The random variable $X$is the number of forcible rapes in Texas every year.

Explanation (part d)

The null hypothesis $H_0$: The proportion of forcible rapes in Texas in $2011and2010$are same.

The alternative hypothesis: $H_a$: The proportion of forcible rapes in Texas in $2011and2010$are different.

The null and alternative hypothesis:

$$\begin{gathered} H_0:p_1=p_2 \\ {H}_a:p_1\ne p_2 \end{gathered}$$

Explanation (part e)

The test is about the difference in the proportion of forcible rapes, therefore, it is two-tailed. Hence, the test is two-tailed.

Explanation (part f)

The p-value is$0.0693$

Explanation (part g)

Here, $p-value>\text{significantvalue,}$so we do not reject the null hypothesis.

Explanation (part h)

At the $5\%$level of significance, from the sample data, there issufficient evidence to conclude that the proportion of forcible rapes in Texas in $2011and2010$are the same.