Buying a Car The following data represent the asking price, in...

Question

Buying a Car The following data represent the asking price, in dollars, for a random sample of 2014 coupes (a two-door car) and a random sample of 2014 Chevy Camaros.

b. Explain why the mean is higher for Camaros yet the standard deviation is less.

Table showing asking prices in dollars for samples of coupes and Camaros with varying values for comparison.

Accepted Answer

All right, hello, everyone. So this question says, a company reports the annual salaries in thousands of dollars for a random sample of 50 senior engineers and a random sample of 50 entry level assistants. The senior engineers have a higher sample mean salary, but a smaller sample standard deviation than the assistants. Which statement best explains this outcome? And here we've got 4 different answer choices labeled A through D. All right, so before we talk about our answer choices, let's first recall the formulas for both sample mean and sample standard deviation. First, let's start with the sample mean or X bar. X bar is. The sum of. All of your data points divided by N, which is the total number of data points that you have. So the sample standard deviation or lowercase s is the square root of One over and subtracted by one. Multiplied by the sum. Of all squared deviations for your data points, and I'll explain what that means in just a moment. OK. So to find the sample standard deviation, you first have to find the sample mean, right? And then from there you subtract each individual data point from the sample mean, and then square that value. That's the squared deviation. So then you take the sum of all of your squared deviations, divide that by and subtracted by 1, and then take the square root of that value to get S. But here we don't have to calculate anything. It's more about understanding the formulas themselves. So, going back to the sample mean formula for a second. It holds That the reason why the senior engineers sample mean was larger was because they had a higher. Sample meat Because their salaries happen to be higher. So going back to that, if X bar is going to be higher for the senior engineers, it follows then. That that's going to have an impact on the sample standard deviation. Now, specifically a higher. Xar tells you that the engineers likely have salaries that are clustered tightly around a higher X bar value. Because those salaries are clustered closer together, the deviation is. Going to be smaller, which means that therefore is going to be smaller, whereas the assistants can oftentimes have more variable salaries. Which in turn produces a larger standard deviation. Ultimately, our correct answer is going to be option B, which reads as follows. The senior engineers's mean is higher and their standard deviation is smaller because standard deviation measures. Average distance from the mean. And engineers salaries are more tightly clustered around a higher mean. And there you have it. So with that being said, thank you so very much for watching and I hope you found this helpful.

Buying a Car The following data represent the asking price, in dollars, for a random sample of 2014 coupes (a two-door car) and a random sample of 2014 Chevy Camaros.
b. Explain why the mean is higher for Camaros yet the standard deviation is less.

Key Concepts

Mean (Average)

Standard Deviation

Data Distribution and Variability

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