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The chi-square test is a statistical method used to determine if there is a significant difference between expected and observed data. It is often employed when you have categorical data and want to assess how well it fits an expected distribution. For our exercise, it's essential to compare the observed frequencies of light truck colors against the expected frequencies to see if they align. The chi-square test statistic is calculated using the formula:

• \[ \chi^2 = \sum \frac{(O_i - E_i)^2}{E_i} \]

Here, \( O_i \) stands for the observed frequency, and \( E_i \) stands for the expected frequency.
The degrees of freedom, which is necessary for finding the critical value, is computed as the number of categories minus one. For this exercise, with seven color categories, the degrees of freedom are six. Using the significance level of \( \alpha = 0.05 \), we compare the computed chi-square statistic against the critical value from chi-square distribution tables. If our test statistic exceeds the critical value, we reject the null hypothesis.

The goodness-of-fit test is a form of chi-square test designed to see how well an observed distribution of data matches a theoretical or expected distribution. In other words, it tests how well the sample data fits a distribution from a broader population.
In our exercise, we use the goodness-of-fit test to check if the observed proportions of truck colors correspond to the expected proportions provided in the problem statement.

• White: 31%
• Black: 19%
• Silver: 11%
• Red: 11%
• Gray: 10%
• Blue: 8%
• Other: 10%

By comparing observed data from our sample of 180 trucks to these expected proportions, we determine if the real-world data fits the hypothetical distribution.
The chi-square statistic is a crucial element in determining the level of goodness-of-fit, as it highlights discrepancies between observed and expected values. If the calculated chi-square statistic is larger than the critical value, the fit is not good, leading us to reject the null hypothesis.

The null hypothesis is a fundamental concept in statistical hypothesis testing. It acts as a starting point or assumption that there is no effect or no difference in the data, which we seek to test.
In the context of the exercise, the null hypothesis states that the distribution of light truck colors does not differ from the expected theoretical proportions:

• \( p_\text{White} = 0.31 \)
• \( p_\text{Black} = 0.19 \)
• \( p_\text{Silver} = 0.11 \)
• \( p_\text{Red} = 0.11 \)
• \( p_\text{Gray} = 0.10 \)
• \( p_\text{Blue} = 0.08 \)
• \( p_\text{Other} = 0.10 \)

This implies that there’s no significant discrepancy between the observed sample data and the expected distribution. The aim of conducting our test is to determine whether to continue believing this hypothesis or to reject it based on evidence from our sample.

The alternative hypothesis is in direct opposition to the null hypothesis and represents what we are trying to establish or prove. When conducting a chi-square test for goodness-of-fit, the alternative hypothesis suggests that the observed distribution of data does not fit the expected distribution.
For this exercise, the alternative hypothesis posits that at least one of the color proportions for light trucks in the stated area is different from the expected proportions:

• For example, the actual proportion of white trucks may not be exactly 31%, as hypothesized.

Accepting the alternative hypothesis indicates a significant deviation between the observed and expected data.
If our test statistic is greater than the critical value, and we reject the null hypothesis, the claim is supported, confirming that differences in truck color proportions exist. This highlights the importance of the alternative hypothesis, as it directs us towards uncovering new insights about the underlying population.