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 = 2.0. For the yellow
paint the mean was x2 = 7.1, with standard
deviation s2 = 2.3. Based on the 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 meansingle proportion difference of meanspaired differencedifference of proportions
What is the level of significance?
State the null and alternate hypotheses.
H0: μ1 = μ2; H1: μ1 ≠ μ2H0: μ1 ≥ μ2; H1: μ1 < μ2 H0: μ1 ≠ μ2; H1: μ1 = μ2H0: μ1 ≤ μ2; H1: μ1 > μ2
What sampling distribution will you use? What assumptions are you
making?
The standard normal. We assume that both population distributions are approximately normal with known population standard deviations.The Student's t. We assume that both population distributions are approximately normal with unknown population standard deviations. The Student's t. We assume that both population distributions are approximately normal with known population standard deviations.The standard normal. We assume that both population distributions are approximately normal with unknown population standard deviations.
What is the value of the sample test statistic? (Test the
difference μ1 − μ2. Round
your answer to three 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 = 0.01 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.
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.
difference of means
level of significance =0.01
H0: μ1 ≤ μ2; H1: μ1 > μ2
the Student's t. We assume that both population distributions are approximately normal with unknown population standard deviations.
std error =√(S21/n1+S22/n2)= | 0.8799 | |
test stat t =(x1-x2-Δo)/Se = | 2.614 |
0.005 < P-value < 0.025
At the α = 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 that the yellow paint has less visibility after 1 year.
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