Question

A study is made of residents in Phoenix and its suburbs concerning the proportion of residents who subscribe to Sporting News. A random sample of 88 urban residents showed that 13 subscribed, and a random sample of 96 suburban residents showed that 20 subscribed. Does this indicate that a higher proportion of suburban residents subscribe to Sporting News? Use a 1% level of significance. What are we testing in this problem? paired difference single proportion difference of proportions difference of means single mean (a) What is the level of significance? .01 State the null and alternate hypotheses. H0: p1 = p2; H1: p1 > p2 H0: p1 = p2; H1: p1 < p2 H0: p1 < p2; H1: p1 = p2 H0: p1 = p2; H1: p1 ? p2 (b) What sampling distribution will you use? What assumptions are you making? The Student's t. We assume the population distributions are approximately normal. 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. The number of trials is sufficiently large. What is the value of the sample test statistic? (Test the difference p1 ? p2. Do not use rounded values. Round your final answer to two decimal places.) (c) Find (or estimate) the P-value. (Round your answer to four decimal places.) Sketch the sampling distribution and show the area corresponding to the P-value. Maple Generated Plot Maple Generated Plot Maple Generated Plot Maple Generated Plot (d) 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 ?? At the ? = 0.01 level, we reject the null hypothesis and conclude the data are statistically significant. At the ? = 0.01 level, we reject the null hypothesis and conclude the data are not statistically significant. At the ? = 0.01 level, we fail to reject the null hypothesis and conclude the data are statistically significant. At the ? = 0.01 level, we fail to reject the null hypothesis and conclude the data are not statistically significant. (e) Interpret your conclusion in the context of the application. There is sufficient evidence at the 0.01 level to conclude that the proportion of suburban residents subscribing to Sporting News is higher. There is insufficient evidence at the 0.01 level to conclude that the proportion of suburban residents subscribing to Sporting News is higher.

Answer 1

The statistical software output for this problem is:

Two sample proportion summary hypothesis test:
p₁ : proportion of successes for population 1
p₂ : proportion of successes for population 2
p₁ - p₂ : Difference in proportions
H₀ : p₁ - p₂ = 0
H_A : p₁ - p₂ < 0

Hypothesis test results:

Difference	Count1	Total1	Count2	Total2	Sample Diff.	Std. Err.	Z-Stat	P-value
p1 - p2	13	88	20	96	-0.060606061	0.056618659	-1.0704256	0.1422

Hence,

Hypotheses:H0: p1 = p2; H1: p1 < p2

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

Test statistic = -1.07

p - Value = 0.1422

At the ? = 0.01 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.

There is insufficient evidence at the 0.01 level to conclude that the proportion of suburban residents subscribing to Sporting News is higher.