emancipezN

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 $x=19.8\text{}and\text{}s=1.5$ . 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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Skilled2020-11-09Added 106 answers

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,$df=34(n\u20131)$ using the t-distribution table.

Therefore, the rejection region is t > 1.691.

The t-statistic can be obtained as:

$t=\frac{\stackrel{\u2015}{x}-{\mu}_{0}}{s}/\sqrt{n}=\frac{19.8-19.6}{1.5}/\sqrt{35}=0.789$

Step 3

Decision: Since, it is observed that$t=0.789<t-critical\text{}value\text{}=1.691$ , 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.

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,

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,

Therefore, the rejection region is t > 1.691.

The t-statistic can be obtained as:

Step 3

Decision: Since, it is observed that

Using the p-value approach: The p-value is

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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