Question

The highway department is testing two types of reflecting paint for concrete bridge end pillars. The two kinds of paint are alike in every respect except that one is orange and the other is yellow. The orange paint is applied to 12 bridges, and the yellow paint is applied to 12 bridges. After a period of 1 year, reflectometer readings were made on all these bridge end pillars. (A higher reading means better visibility.) For the orange paint, the mean reflectometer reading was x₁ = 9.4, with standard deviation s₁ = 2.1. For the yellow paint the mean was x₂ = 6.9, with standard deviation s₂ = 2.4. Based on these data, can we conclude that the yellow paint has less visibility after 1 year? Use a 1% level of significance.

What are we testing in this problem?

paired difference

single proportion    

single mean

difference of means

difference of proportions

(a) What is the level of significance?

State the null and alternate hypotheses.

H₀: μ₁ = μ₂; H₁: μ₁ > μ₂

H₀: μ₁ = μ₂; H₁: μ₁ ≠ μ₂    

H₀: μ₁ = μ₂; H₁: μ₁ < μ₂

H₀: μ₁ > μ₂; H₁: μ₁ = μ₂

(b) What sampling distribution will you use? What assumptions are you making?

The Student's t. We assume that both population distributions are approximately normal with unknown standard deviations.

The Student's t. We assume that both population distributions are approximately normal with known standard deviations.   

The standard normal. We assume that both population distributions are approximately normal with known standard deviations.

The standard normal. We assume that both population distributions are approximately normal with unknown standard deviations.

What is the value of the sample test statistic? (Test the difference μ₁ − μ₂. Round your answer to three decimal places.)

(c) Find (or estimate) the P-value.

P-value > 0.250

0.125 < P-value < 0.250   

0.050 < P-value < 0.125

0.025 < P-value < 0.050

0.005 < P-value < 0.025

P-value < 0.005

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

(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 α?

At the α = 0.01 level, we reject the null hypothesis and conclude the data are statistically significant.

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

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

At the α = 0.01 level, we fail to reject the null hypothesis and conclude the data are statistically significant.
(e) Interpret your conclusion in the context of the application.

There is sufficient evidence at the 0.01 level that the yellow paint has less visibility after 1 year.

There is insufficient evidence at the 0.01 level that the yellow paint has less visibility after 1 year.   

Answer 1

Difference of mean test.

a)

Level of significance (alpha) = 1% = 0.01

H0 :

Ha :

b)

The Student's t. We assume that both population distributions are approximately normal with unknown standard deviations.

Test statistic :-

t = -2.716

c)

P-value :-

P = 0.0064

0.005 < p-value < 0.025

d)

At the α = 0.01 level, we reject the null hypothesis and conclude the data are statistically significant.

e)

There is sufficient evidence at the 0.01 level that the yellow paint has less visibility after 1 year.