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 11 bridges, and the yellow paint is applied to 11 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 = 6.4, with standard deviation s2 = 1.7. 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? single mean single proportion paired difference 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 standard normal. 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 unknown standard deviations. The standard normal. We assume that both population distributions are approximately normal with known standard deviations. The Student's t. We assume that both population distributions are approximately normal with known standard deviations. What is the value of the sample test statistic? (Test the difference μ1 − μ2. Round your answer to three decimal places.) (d) 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 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

Given that,
mean(x)=9.4
standard deviation , s.d1=2.1
number(n1)=11
y(mean)=6.4
standard deviation, s.d2 =1.7
number(n2)=11
null, Ho: u1 = u2
alternate, H1: u1 > u2
level of significance, α = 0.01
from standard normal table,right tailed t α/2 =2.764
since our test is right-tailed
reject Ho, if to > 2.764
we use test statistic (t) = (x-y)/sqrt(s.d1^2/n1)+(s.d2^2/n2)
to =9.4-6.4/sqrt((4.41/11)+(2.89/11))
to =3.6826
| to | =3.6826
critical value
the value of |t α| with min (n1-1, n2-1) i.e 10 d.f is 2.764
we got |to| = 3.68261 & | t α | = 2.764
make decision
hence value of | to | > | t α| and here we reject Ho
p-value:right tail - Ha : ( p > 3.6826 ) = 0.00211
hence value of p0.01 > 0.00211,here we reject Ho
ANSWERS
---------------
a.
null, Ho: u1 = u2
alternate, H1: u1 > u2
b.
The standard normal. We assume that both population distributions are approximately normal with known standard deviations.
The Student's t
c.
test statistic: 3.682
critical value: 2.764
decision: reject Ho
p-value: 0.0021
d.
At the α = 0.01 level, we reject the null hypothesis and conclude the data are statistically significant
e.
we have enough evidence to support the claim that the yellow paint has less visibility after 1 year