Question

The Pew Research Group conducted a poll in which they asked, "Are you in favor of, or opposed to, executing persons as a general policy when the crime was committed while under the age of 18?" Of the 580 Catholics surveyed, 180 indicated they favored capital punishment; of the 600 scculars (those who do not associate with a religion) surveyed, 238 favored capital punishment. Is there a significant difference in the proportion of individuals in these groups in favor of capital punishment for persons under the age of 18? Use the ae 0.01 level of significance.

Answer 1

Answer:

yes

Step-by-step explanation:

Given:  

x_1= 180

n_1= 580

x_2= 238

n_2= 600

   $\alpha$= 0.01

Given claim: difference  

The claim is either the null hypothesis or the alternative hypothesis. The null hypothesis states that the population proportion is equal to the value mentioned in the claim. If the null hypothesis is the claim, then the alternative hypothesis states the opposite of the null hypothesis.  

H_0:p_1=p_2

H_0:p_1$\neq$p_2

CRITICAL VALUE APPROACH  

The sample proportion is the number of successes divided by the sample size:  

p_1=x_1/n_1=0.3103

p_2=x_2/n_2=0.40

p_p=(x_1+x_2)/(n_1+n_2) = 0.36

Determine the value of the test statistic:  

z=(p_1-p_2) ÷ (√p_p(1-p_p)*√(1/n_1+1/n_2)

 =-3.10

The critical values are the values corresponding to a probability of $\alpha$/2 = 0.025 and 1-$\alpha$/2 = 0.975 in the normal probability table in the appendix:  

z = ±2.575  

The rejection region then contains all values below —2.575 and all values above 2.575.  

If the value of the test statistic is within the rejection region, then the null hypothesis is rejected:  

—3.10 < —2.575 ==> Reject H_0

There is sufficient evidence to support the claim that there is a difference in  the proportion of individuals in these in favor of capital punishment  for persons under the age of 18.  

P-VALUE APPROACH  

The sample proportion is the number of successes divided by the sample size:  

p_1=x_1/n_1=0.3103

p_2=x_2/n_2=0.40

p_p=(x_1+x_2)/(n_1+n_2) = 0.36

Determine the value of the test statistic:  

z=(p_1-p_2) ÷ (√p_p(1-p_p)*√(1/n_1+1/n_2)

 =-3.10

The P-value is the probability of obtaining the value of the test statistic, or a value more extreme, assuming that the null hypothesis is true. Determine the P-value using table the normal probability table in the appendix:  

P = P(Z < –3.10 or Z > 3.10) = 2P(Z < –3.10) = 2(0.0010) = 0.0020  

If the P-value is smaller than the significance level, then reject the null hypothesis:  

P < 0.01 ==> Reject H_0

There is sufficient evidence to support the claim that there is a difference in the proportion of individuals in these groups in for of capital punishment for persons under the age of 18.