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A t-test is a statistical test used to determine if there is a significant difference between the means of two groups. It is particularly useful when working with small sample sizes and when the population standard deviation is unknown. In our context, the t-test helps us assess whether the mean reflectometer reading (\(\mu\)) for a new type of road paint is greater than 20, which would indicate that the paint reflects enough light to be clearly visible at night.

The test involves calculating a test statistic (in this case, the provided t-value of 3.2) from the collected data and comparing it to a critical value from the t-distribution table. The t-test can be one-tailed or two-tailed, depending on the alternative hypothesis. For this exercise, the one-tailed t-test is used because the alternative hypothesis (\(H_{a}: \mu > 20\)) suggests a directional change in the mean.

The steps in conducting a t-test include:

• Calculating the t-statistic based on the difference between the sample mean and the hypothesized population mean, scaled by standard error.
• Determining the degrees of freedom, which is the sample size minus one.
• Using the t-distribution table to find the critical value for the chosen significance level.
• Comparing the calculated t-statistic to the critical value to decide whether to reject the null hypothesis.

The significance level, often denoted by \(\alpha\), is a critical threshold used in hypothesis testing. It represents the probability of rejecting the null hypothesis when it is actually true, which is also known as Type I error.

In this exercise, different significance levels are evaluated: 0.05, 0.01, and 0.001. The choice of \(\alpha\) affects the critical values used for comparison with the test statistic.

• For \(\alpha = 0.05\), the test is considered moderately strict; it means there's a 5% risk of concluding that a difference exists when there is none.
• For \(\alpha = 0.01\), the test is stricter, decreasing the risk of Type I error to 1%.
• For \(\alpha = 0.001\), the test is very strict, allowing only a 0.1% probability of mistakenly rejecting the null hypothesis.

These levels determine how conservatively the hypotheses are tested and which critical t-values the t-statistic needs to exceed in order for the null hypothesis to be rejected. A lower significance level means more evidence is required to reject \(H_{0}\).

Degrees of freedom (df) in the context of the t-test refer to the number of values in the final calculation of a statistic that are free to vary. Understanding degrees of freedom is crucial for interpreting the results of the t-test because they influence the shape of the t-distribution used to find critical values.

In this exercise, we have a sample size (\(n\)) of 15 observations. The degrees of freedom are calculated as \(n - 1\), which means 14 in this case. The t-distribution with 14 degrees of freedom will be used to determine the critical t-values for each significance level, ensuring that we have the correct threshold for deciding whether to reject the null hypothesis.

As the sample size changes, the degrees of freedom also change, which can impact the results of the test.

• With more degrees of freedom (larger sample), the t-distribution approaches a normal distribution.
• With fewer degrees of freedom, the tails of the distribution are heavier, which means we need a larger t-value to reject \(H_0\).

These considerations highlight the importance of calculating degrees of freedom accurately to correctly compare the test statistic against critical values.