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2. (6 pts) In a recent year, the most popular colors for light trucks were white (30%), black (18%), silver (12%), red (12%), gray (10%), blue (8%), and other (10%).

A survey of randomly selected light truck owners in a particular area revealed the following:

\begin{tabular}{lllllll}
White & Black & Silver & Red & Gray & Blue & Other \\
55 & 40 & 26 & 25 & 22 & 20 & 12
\end{tabular}

At [tex]$\alpha = 0.05$[/tex], do the proportions differ from those stated? Test the claim that the percentages are equal.



Answer :

GinnyAnswer
Certainly! Let's break down the process to test the claim that the proportions of light truck colors in a particular area match the stated proportions.

### Step 1: State the Hypotheses
- Null Hypothesis ([tex]\( H_0 \)[/tex]): The observed proportions match the stated proportions.
- Alternative Hypothesis ([tex]\( H_a \)[/tex]): The observed proportions do not match the stated proportions.

### Step 2: Observed and Expected Frequencies
From the problem, the observed frequencies for the various colors are:
- White: 55
- Black: 40
- Silver: 26
- Red: 25
- Gray: 22
- Blue: 20
- Other: 12

The stated proportions for these colors are:
- White: 30%
- Black: 18%
- Silver: 12%
- Red: 12%
- Gray: 10%
- Blue: 8%
- Other: 10%

### Step 3: Calculate the Expected Frequencies
To find the expected frequencies, we need to multiply the total number of observations by the expected proportions. Let [tex]\( N \)[/tex] be the total number of observed frequencies.

[tex]\[ N = 55 + 40 + 26 + 25 + 22 + 20 + 12 = 200 \][/tex]

Now, we calculate the expected frequencies for each color:
- White: [tex]\( 0.30 \times 200 = 60 \)[/tex]
- Black: [tex]\( 0.18 \times 200 = 36 \)[/tex]
- Silver: [tex]\( 0.12 \times 200 = 24 \)[/tex]
- Red: [tex]\( 0.12 \times 200 = 24 \)[/tex]
- Gray: [tex]\( 0.10 \times 200 = 20 \)[/tex]
- Blue: [tex]\( 0.08 \times 200 = 16 \)[/tex]
- Other: [tex]\( 0.10 \times 200 = 20 \)[/tex]

### Step 4: Perform the Chi-Square Test
The chi-square test statistic is calculated using the formula:

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

Where [tex]\( O_i \)[/tex] are the observed frequencies and [tex]\( E_i \)[/tex] are the expected frequencies.

Given the chi-square test statistic and the p-value:
[tex]\[ \chi^2 = 5.469 \][/tex]
[tex]\[ p\text{-value} = 0.485 \][/tex]

### Step 5: Conclusion
We compare the p-value to the significance level ([tex]\( \alpha = 0.05 \)[/tex]).

- If the [tex]\( p\text{-value} \leq \alpha \)[/tex], we reject the null hypothesis.
- If the [tex]\( p\text{-value} > \alpha \)[/tex], we do not reject the null hypothesis.

Given that the [tex]\( p\text{-value} = 0.485 \)[/tex] is greater than [tex]\( \alpha = 0.05 \)[/tex], we do not reject the null hypothesis. There is insufficient evidence to conclude that the proportions of light truck colors in this particular area differ from those stated.

### Final Conclusion
At the [tex]\( \alpha = 0.05 \)[/tex] significance level, there is not enough evidence to reject the null hypothesis. Hence, we conclude that the proportions of light truck colors in this particular area are not significantly different from the stated proportions.

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