Question

Each model year seems to introduce new colors and different hues for a wide array of vehicles, from luxury cars, to full-size or intermediate models, to compacts and sports cars, to light trucks. However, white and silver/gray continue to make the top five or six colors across all of these categories of vehicles. The top six colors and their percentage of the market share for compact/sports cars are shown in the following table. To verify the figures, a random sample consisting of 250 compact/sports cars was taken and the color of the vehicles recorded. The sample provided the following counts for the categories given above: \(52,43,48,41,\) \(32,19,\) respectively. a. Is any category missing in the classification? How many vehicles belong to that category? b. Is there sufficient evidence to indicate that our percentages of the colors for compact/sports cars differ from those given? Find the approximate \(p\) -value for the test.

Short answer

Answer: No, there is not a significant difference between the given percentages of compact/sports car colors and the observed sample data.

Step-by-step solution

Calculate Total Observed Vehicles and Missing Category

Count the total number of vehicles in the given sample:\(52+43+48+41+32+19=235\). The sample size is 250, so there is a missing category with \(250-235=15\) vehicles.

State the Null and Alternative Hypotheses

Null hypothesis(\(H_0\)): The given percentages of colors for compact/sports cars match the sample proportions. Alternative hypothesis(\(H_1\)): The given percentages of colors for compact/sports cars differ from the sample proportions.

Calculate Expected Count for Each Category

Use the given market share percentages to calculate the expected count for each category, assuming the null hypothesis is true: White: \((0.22)(250) = 55\), Silver/Gray: \((0.18)(250) = 45\), Blue: \((0.21)(250) = 52.5\), Red: \((0.17)(250) = 42.5\), Black: \((0.12)(250) = 30\), Other: \((0.07)(250) = 17.5\)

Calculate the Chi-Square Statistic

Calculate the Chi-Square test statistic using the formula: $$\chi^2 = \sum_{i=1}^n \frac{(O_i-E_i)^2}{E_i}$$ Here \(O_i\) represents the observed count and \(E_i\) represents the expected count for each category. $$\chi^2 = \frac{(52-55)^2}{55} + \frac{(43-45)^2}{45} + \frac{(48-52.5)^2}{52.5} + \frac{(41-42.5)^2}{42.5} + \frac{(32-30)^2}{30} + \frac{(19-17.5)^2}{17.5}$$ Calculate the Chi-Square test statistic: $$\chi^2 = 0.164+0.089+0.387+0.053+0.133+0.128 = 0.954$$

Find the p-value

Using a Chi-Square distribution table or calculator, determine the p-value corresponding to a test statistic of 0.954 with 5 degrees of freedom (6 categories minus 1). The p-value is approximately 0.96.

Interpret the Results and Draw Conclusions

Compare the p-value (0.96) to a significance level, usually \(\alpha=0.05\). Since the p-value is greater than \(\alpha\), we fail to reject the null hypothesis. Therefore, there is not sufficient evidence to indicate that the given percentages of colors for compact/sports cars differ from those observed.

Key concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to determine whether there is enough evidence in a sample of data to infer that a particular condition is true for the entire population. The process begins by formulating two opposite hypotheses:

• Null Hypothesis ( $H_0$): This is a statement that there is no effect or no difference. It represents the status quo or a baseline assumption. In our case, it assumes that the given percentages of compact/sports car colors accurately represent the market.
• Alternative Hypothesis ( $H_1$): This is the statement that we are trying to find evidence for. It suggests that there is a difference or effect. Here, it suggests that the observed color distribution differs from the expected percentages.

The hypothesis test runs through several steps: calculation of the test statistic (such as chi-square), determining the p-value, and interpreting these in the context of the significance level.
If the p-value is less than the chosen significance level (often 0.05), the null hypothesis is rejected, indicating that the observed data is significantly different from what was expected under the null hypothesis.

Market Share Analysis

Market share analysis involves identifying the portion of a market controlled by a particular company, product, or in this case, a color of compact/sports cars. Here, the market share is expressed as percentages for each car color category, indicating how dominant each color is within the market.
The analysis assists businesses in understanding their market position and strategizing accordingly. With a given data, it becomes easier to see whether the predominance of certain colors has changed over time or remains fairly constant.

• Expected Counts: Using the market share percentage, you can estimate how many cars in a sample should be a specific color if the market share holds true. For example, if the market share for blue cars is 21%, and your sample size is 250, then you would expect about 52.5 cars in the sample to be blue.
• Purpose: It helps in comparing observed counts with expected counts to perform hypothesis testing. This can inform researchers whether market trends predicted by companies are realistic or need adjustment.

Understanding the market share gives insight into consumer preferences and guides future decisions in production and marketing strategies.

Compact Cars Color Distribution

Compact cars and sports cars often come in a variety of colors appealing to different consumer tastes. Analyzing the color distribution involves studying how often each color appears in a given sample of these cars.
To examine this distribution properly, a researcher must collect a sample—such as 250 compact/sports cars—and observe the frequency of each color. This data is then compared against expected market percentages to determine if preferences are changing.
In the exercise, we check whether the distribution of car colors in our sample matches the expected percentages. Each color (like white, silver/gray, blue, etc.) has its market share percentage, giving us a target to compare against.

• Importance: Understanding color distribution allows manufacturers to better align their production with what the market demands, ensuring they meet consumer preferences effectively.
• Chi-Square Statistic: This statistic quantifies the discrepancy between observed and expected counts for each color. A small chi-square value suggests that the observed distribution aligns well with expected values, implying consistent consumer preference.

Through examining such distributions, automakers can adjust production strategies to maintain or increase profitability.