Suppose a Normal model describes the fuel efficiency of cars currently registered in your state. The mean is 27 mpg, with a standard deviation of 4 mpg. Describe the mileage of the most efficient 3% of all cars.
a. greater than 24.9 mpg
b. greater than 19.5 mpg
c. greater than 34.5 mpg
d. less than 19.5 mpg
Your Z table tells you that about 97% of the population lies below the mean + 1.88 std.
so, ...
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To describe the mileage of the most efficient 3% of all cars, we need to find the cutoff value that separates the top 3% from the rest of the data. This cutoff value is typically referred to as the "z-score," which represents how many standard deviations away from the mean a particular value is.
In this case, we are given that the mean of the fuel efficiency is 27 mpg, with a standard deviation of 4 mpg. We want to find the value that corresponds to the top 3% of the distribution.
To find this value, we can use the Z-table or a statistical calculator. The Z-table provides the area under the standard normal distribution curve, which tells us the probability of obtaining a value less than or equal to a given Z-score. Since we are interested in the top 3%, we need to find the Z-score that corresponds to a cumulative probability of 0.97 (100% - 3%).
Using the Z-table or a calculator, we find that the Z-score corresponding to a cumulative probability of 0.97 is approximately 1.88.
To calculate the mileage cutoff, we can use the formula:
Cutoff value = Mean + (Z-score * Standard Deviation)
Substituting the values into the formula:
Cutoff value = 27 + (1.88 * 4)
Cutoff value ≈ 34.52
Therefore, the mileage of the most efficient 3% of all cars is greater than 34.52 mpg.
Hence, the correct answer is c. greater than 34.5 mpg.