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

n₁ = 88

urban residents showed that

10

subscribed, and a random sample of

n₂ = 97

suburban residents showed that

19

subscribed. Does this indicate that a higher proportion of suburban residents subscribe to Sporting News? Use a 5% level of significance.

What are we testing in this problem?

paired differencedifference of means    single proportiondifference of proportionssingle mean

What is the level of significance?


State the null and alternate hypotheses.

H₀: p₁ ≥ p₂; H₁: p₁ < p₂H₀: p₁ > p₂; H₁: p₁ ≤ p₂    H₀: p₁ ≠ p₂; H₁: p₁ = p₂H₀: p₁ ≤ p₂; H₁: p₁ > p₂H₀: p₁ < p₂; H₁: p₁ ≥ p₂H₀: p₁ = p₂; H₁: p₁ ≠ p₂

What sampling distribution will you use? What assumptions are you making?

The standard normal. The number of trials is sufficiently large.The Student's t. 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.

What is the value of the sample test statistic? (Test the difference p₁ − p₂). Round your answer to two decimal places.


Estimate the P-value.

P-value > 0.2500.125 < P-value < 0.250    0.050 < P-value < 0.1250.025 < P-value < 0.0500.005 < P-value < 0.025P-value < 0.005

Sketch the sampling distribution and show the area corresponding to the P-value.

Will you reject or fail to reject the null hypothesis? Are the data statistically significant at level α?

At the α = 0.05 level, we reject the null hypothesis and conclude the data are statistically significant.At the α = 0.05 level, we reject the null hypothesis and conclude the data are not statistically significant.    At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are statistically significant.At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.

Interpret your conclusion in the context of the application.

There is sufficient evidence at the 0.05 level to conclude that the proportion of suburban residents subscribing to Sporting News is higher.There is insufficient evidence at the 0.05 level to conclude that the proportion of suburban residents subscribing to Sporting News is higher.    

Answer 1

Given that,

For urban residents : n1 = 88, x1 = 10 and

For suburban : n2 = 97, x2 = 19 and

Pooled proportion is,

This is difference of proportion test. (Two-proportion test).

Level of significance = 0.05

The null and alternative hypotheses are,

H0 : p1 ≥ p2

Ha : p1 < p2

The sampling distribution of the sample proportions is The standard normal. The number of trials is sufficiently large.

Test statistic is,

=> Test statistic = Z = -1.54

p-value = P(Z < -1.54) = 0.0618

0.050 < P-value < 0.125

Since, p-value is greater than 0.05, we fail to reject H0.

At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.

​​​Conclusion : There is insufficient evidence at the 0.05 level to conclude that the proportion of suburban residents subscribing to Sporting News is higher.  

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