2020-12-07

Would you rather spend more federal taxes on art? Of a random sample of ${n}_{1}=86$ politically conservative voters, ${r}_{1}=18$ responded yes. Another random sample of ${n}_{2}=85$ politically moderate voters showed that ${r}_{2}=21$ responded yes. Does this information indicate that the population proportion of conservative voters inclined to spend more federal tax money on funding the arts is less than the proportion of moderate voters so inclined? Use $\alpha =0.05.$

a) State the null and alternate hypotheses.

${H}_{0}:{p}_{1}={p}_{2},{H}_{1}:{p}_{1}>{p}_{2}$

${H}_{0}:{p}_{1}={p}_{2},{H}_{1}:{p}_{1}<{p}_{2}$

${H}_{0}:{p}_{1}={p}_{2},{H}_{1}:{p}_{1}\ne {p}_{2}$

${H}_{0}:{p}_{1}<{p}_{2},{H}_{1}:{p}_{1}={p}_{2}$

b) What sampling distribution will you use? What assumptions are you making? The Student's t. The number of trials is sufficiently large. The standard normal. The number of trials is sufficiently large.The standard normal. We assume the population distributions are approximately normal. The Student's t. We assume the population distributions are approximately normal.

c)What is the value of the sample test statistic? (Test the difference ${p}_{1}-{p}_{2}$. Do not use rounded values. Round your final answer to two decimal places.)

d) Find (or estimate) the P-value. (Round your answer to four decimal places.)

e) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level alpha? At the $\alpha =0.05$ level, we reject the null hypothesis and conclude the data are statistically significant. At the $\alpha =0.05$ level, we fail to reject the null hypothesis and conclude the data are statistically significant. At the $\alpha =0.05$ level, we fail to reject the null hypothesis and conclude the data are not statistically significant. At the $\alpha =0.05$ level, we reject the null hypothesis and conclude the data are not statistically significant.

f) Interpret your conclusion in the context of the application. Reject the null hypothesis, there is sufficient evidence that the proportion of conservative voters favoring more tax dollars for the arts is less than the proportion of moderate voters. Fail to reject the null hypothesis, there is sufficient evidence that the proportion of conservative voters favoring more tax dollars for the arts is less than the proportion of moderate voters. Fail to reject the null hypothesis, there is insufficient evidence that the proportion of conservative voters favoring more tax dollars for the arts is less than the proportion of moderate voters. Reject the null hypothesis, there is insufficient evidence that the proportion of conservative voters favoring more tax dollars for the arts is less than the proportion of moderate voters.

Part a

State the hypotheses. Let ${p}_{1}$ denotes the population proportion of conservative voters. Let ${p}_{2}$ denotes the population proportion of moderate voters. Null hypothesis: ${H}_{0}:{p}_{1}-{p}_{2}=0$ That is, there is no evidence to conclude that the population proportion of conservative voters inclined to spend more federal tax money on funding the arts is less than the proportion of moderate voters. Alternative hypothesis: ${H}_{1}:{p}_{1}-{p}_{2}<0$

That is, there is evidence to conclude that the population proportion of conservative voters inclined to spend more federal tax money on funding the arts is less than the proportion of moderate voters.

Part b

Determine the assumptions that required for two sample proportions: Conditions for using the z - procedure for the difference between two proportions: The data collected for the samples must be selected randomly. Each sample must contain at least 10 success, that is $n\stackrel{^}{p}\ge 10.$ Each sample must contain at least 10 failures, that is $n\left(1-\stackrel{^}{p}\right)\ge 10.$ Each observation in the sample must be independent of each other. From the information, given that ${r}_{1}=18,{n}_{1}=86$
${r}_{2}=21,{n}_{2}=85$

Condition 1: ${n}_{1}{\stackrel{^}{p}}_{1}=86\left(\frac{18}{86}\right)$
$=18$
$\ge 10$
${n}_{1}\left(1-{\stackrel{^}{p}}_{1}\right)=86\left(1-\frac{18}{86}\right)$
$=86×\frac{68}{86}$
$=68$
$\ge 10$

Condition 2: ${n}_{2}{\stackrel{^}{p}}_{2}=85\left(\frac{21}{85}\right)$
$=21$
$\ge 10$
${n}_{2}\left(1-{\stackrel{^}{p}}_{2}\right)=85\left(1-\frac{21}{85}\right)$
$=85×\left(\frac{64}{85}\right)$
$=64$
$\ge 10$

Moreover the two samples are randomly selected and are independent to each other. Since, all the conditions have been satisfied the listed assumptions are turned to be realistic The standard normal. We assume the population distributions are approximately normal.

Answer: The standard normal. We assume the population distributions are approximately normal.

Part c

Obtain the value of the test statistic. The value of test statistic is obtained below: The value of sample proportion 1, ${\stackrel{^}{p}}_{1}=\frac{{x}_{1}}{{n}_{1}}$
$=\frac{18}{86}$
$=0.2093$

The value of sample proportion 2, ${\stackrel{^}{p}}_{2}=\frac{{x}_{2}}{{n}_{2}}$
$=\frac{21}{85}$
$=0.2471$

The value of hat p is, $\stackrel{^}{p}=\frac{{x}_{1}+{x}_{2}}{{n}_{1}+{n}_{2}}$
$=\frac{18+21}{86+85}=\frac{39}{171}$
$=0.2281$

The required value of test statistic is, $z=\frac{{\stackrel{^}{p}}_{1}{\stackrel{^}{p}}_{2}}{\sqrt{\stackrel{^}{p}\left(1-\stackrel{^}{p}\right)}\sqrt{\frac{1}{{n}_{1}}+\frac{1}{{n}_{2}}}}$
$=\frac{0.2471-0.2281}{\sqrt{0.2281\left(1-0.2281\right)}\sqrt{\frac{1}{86}+\frac{1}{85}}}$
$=\frac{0.0190}{0.4196×0.1529}$
$=\frac{0.0190}{0.0642}$
$=0.30$

Thus, the value of test statistic is 0.30.

Part d

Obtain the P-value. Use Excel to obtain the probability value for $Z=0.30$ Follow the instruction to obtain the P-value: 1. Open EXCEL 2. Go to Formula bar. 3. In formula bar enter the function as "=NORMSDIST" 4. Enter the test statistic Z as 0.30 5. Click enter EXCEL output: From the Excel output, the P-value is 0.6179 Thus, the P-value is 0.6179.

Part e

The conclusion is obtained as shown below: Use the level of significance is 0.05. The p-value is greater than level of significance. That is: p-value $\left(=0.6179\right)>\alpha \left(=0.05\right)$ By rejection rule, do not reject the null hypothesis.

Part f

Final conclusion: Fail to reject the null hypothesis, there is insufficient evidence that the proportion of conservative voters favoring more tax dollars for the arts is less than the proportion of moderate voters.