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 *x*_{1} = 9.4, with
standard deviation *s*_{1} = 2.0. For the yellow
paint the mean was *x*_{2} = 7.1, with standard
deviation *s*_{2} = 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.

*H*_{0}: *μ*_{1} =
*μ*_{2}; *H*_{1}:
*μ*_{1} ≠
*μ*_{2}*H*_{0}:
*μ*_{1} ≥ *μ*_{2};
*H*_{1}: *μ*_{1} <
*μ*_{2} *H*_{0}:
*μ*_{1} ≠ *μ*_{2};
*H*_{1}: *μ*_{1} =
*μ*_{2}*H*_{0}:
*μ*_{1} ≤ *μ*_{2};
*H*_{1}: *μ*_{1} >
*μ*_{2}

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 *μ*_{1} − *μ*_{2}. 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 < 0.025*P*-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.

Answer #1

difference of means

level of significance =0.01

*H*_{0}: *μ*_{1} ≤
*μ*_{2}; *H*_{1}:
*μ*_{1} > *μ*_{2}

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

std error
=√(S^{2}_{1}/n_{1}+S^{2}_{2}/n_{2})= |
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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for concrete bridge end pillars. The two kinds of paint are alike
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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...

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