Question

Paint used to paint lines on roads must reflect enough light to be clearly visible at night. Let \(\mu\) denote the true average reflectometer reading for a new type of paint under consideration. A test of \(H_{0}: \mu=20\) versus \(H_{a}: \mu>20\) based on a sample of 15 observations gave \(t=3.2\). What conclusion is appropriate at each of the following significance levels? a. \(\alpha=.05\) c. \(\alpha=.001\) b. \(\alpha=.01\)

Short answer

The actual conclusion depends on the critical t-values for each significance level. Generally, the value \(t=3.2\) would be compared to the critical t-value for each significance level. If \(t=3.2\) is greater than the critical t-value, then the null hypothesis \(H_{0}: \mu=20\) would be rejected, leading to the conclusion that the true average reflectometer reading for the new type of paint is more than 20.

Step-by-step solution

Determine the critical t-values

Consult a t-table to find the critical t-values corresponding to the calculated degrees of freedom, which is n-1 = 15-1 = 14, for each of the given significance levels. The critical t-values are critical because they form a threshold: if the calculated t-value is greater than the critical t-value, then the null hypothesis should be rejected.

Compare the calculated t-value to the critical t-values

The calculated t-value in this case is 3.2. Now this needs to be compared with each of the critical t-values determined in Step 1.

Draw conclusions for each significance level

For each significance level, if the calculated t-value is greater than the critical t-value, then the null hypothesis should be rejected, suggesting that the true average reflectometer reading for the new type of paint is more than 20. If not, the null hypothesis cannot be rejected.