Questions: We reject the null hypothesis and conclude that there is sufficient evidence at a 0.10 level of significance to support the claim that the average time required to install the tile differs from 9 hours.

We reject the null hypothesis and conclude that there is sufficient evidence at a 0.10 level of significance to support the claim that the average time required to install the tile differs from 9 hours.

Transcript text: We reject the null hypothesis and conclude that there is sufficient evidence at a 0.10 level of significance to support the claim that the average time required to install the tile differs from 9 hours.

Solution

Solution Steps

Step 1: State the Null and Alternative Hypotheses

The null hypothesis $H_0$: $\mu = \mu_0$, where $\mu$ is the population mean and $\mu_0$ is the reported or standard time. The alternative hypothesis $H_a$: $\mu \neq \mu_0$.

Step 2: Calculate the Test Statistic

Using the formula for the z-score: $Z = \frac{\bar{x} - \mu_0}{\frac{s}{\sqrt{n}}}$, where $\bar{x} = 8.1$, $\mu_0 = 9$, $s = 3.3$, and $n = 54$, we find that the z-score is $Z = -2$.

Step 3: Determine the Critical Value(s)

Given a significance level of $\alpha = 0.1$, the critical values from the standard normal distribution are approximately $\pm1.64$.

Step 4: Make a Decision

Since the absolute value of the test statistic is greater than the critical value, we reject the null hypothesis. This means there is sufficient evidence to say the average time is significantly different from the reported time.