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2020-11-08

Replacement of paint on highways and streets represents a large investment of funds by state and local governments each year. A new, cheaper brand of paint is tested for durability after one month’s time by reflectometer readings. For the new brand to be acceptable, it must have a mean reflectometer reading greater than 19.6. The sample data, based on 35 randomly selected readings, show . Do the sample data provide sufficient evidence to conclude that the new brand is acceptable? Conduct hypothesis test using $a=.05$. Use the traditional approach and the p-value approach to hypothesis testing! Show all of the steps of the hypothesis test for each approach.

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Step 1
According to the provided data, the sample mean is 19.8 and the sample standard deviation is 1.5 and the sample size is 35.
The null and alternative hypotheses are,
${H}_{0}:\mu =19.6$
${H}_{a}:\mu >19.6$
This corresponds to a right tailed test.
Step 2
The level of significance is 0.05 and the critical value is obtained as 1.691 for degrees of freedom, $df=34\left(n–1\right)$ using the t-distribution table.
Therefore, the rejection region is t > 1.691.
The t-statistic can be obtained as:
$t=\frac{\stackrel{―}{x}-{\mu }_{0}}{s}/\sqrt{n}=\frac{19.8-19.6}{1.5}/\sqrt{35}=0.789$
Step 3
Decision: Since, it is observed that , therefore, the null hypothesis is failed to be rejected.
Using the p-value approach: The p-value is $p=0.2178$ using the standard normal table. Since the $p-value=0.2178$ is greater than the level of significance = 0.05. The null hypothesis is failed to be rejected.
Therefore, it can be concluded that there is not enough evidence to claim that the mean reflectometer reading greater than 19.6.

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