Question # 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

ANSWERED 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 $$x =19.8\ and\ s=1.5$$. 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. 2020-11-09
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, $$df = 34 (n – 1)$$ using the t-distribution table.
Therefore, the rejection region is t > 1.691.
The t-statistic can be obtained as:
$$t=\frac{\overline{x}-\mu_{0}}{s}/\sqrt{n}=\frac{19.8-19.6}{1.5}/\sqrt{35}=0.789$$
Step 3
Decision: Since, it is observed that $$t = 0.789 < t-critical\ value\ = 1.691$$, 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.