1. Gas Mileage Problem
The EPA efficiency rating Ina particular model of new car says that the car is expected to get 22.7 miles per gallon (mpg) of gasoline. Assume that the standard deviation is 1.4 mpg and that the actual mileage are normally distributed about 22.7. The factory produces 2000 of this model.
a. How many of the cars would be expected to get above 23 mpg?
My answer: 830 cars
b. How many would be expected to get below 20 mpg?
My answer: 54 cars
c. How many would be expected to get mileages in the interval from 21 to 24 mpg?
My answer: 1422 cars
d. In what interval about 22.7 could you expect 80% of the mileage to be?
My answer: between 20.9 and 24.5
e. If you purchase one of the cars at random, what is the probability that it will get at least 21.5 mpg?
My answer: 80.4%
f. Suppose you purchased a car of this model and it got only 17.2 mpg. If dealer's service manager told you that this low a mileage was normal for this model car, how would you refute his claim based on statistic? I am stunt on this!
To answer question f, you can use statistical reasoning to refute the claim that a mileage of 17.2 mpg is normal for this model car.
First, you need to understand the concept of standard deviation. In this problem, the standard deviation is given as 1.4 mpg. This value represents the average amount of variation or deviation from the mean (22.7 mpg).
To determine if a value is within a normal range, you can look at how many standard deviations it is away from the mean. Generally, values within two standard deviations of the mean are considered normal or typical.
To calculate the number of standard deviations a value is away from the mean, you can use the following formula:
Standard Deviations (z) = (Value - Mean) / Standard Deviation
For the given value of 17.2 mpg, the calculation would be:
z = (17.2 - 22.7) / 1.4 ≈ -3.93
This means that a mileage of 17.2 mpg is nearly 3.93 standard deviations below the mean.
Next, you can use a statistical table (such as the Z-table or a calculator) to find the probability associated with this z-score. The probability represents the likelihood of observing a value as extreme or more extreme than the given value under the assumption of a normal distribution.
Using the Z-table, you can find that the probability of a value being 3.93 standard deviations or more below the mean is extremely low. It is approximately 0.0001 or 0.01%.
Therefore, based on statistics, you can refute the claim that a mileage of 17.2 mpg is normal for this model car. The probability of observing such a low mileage is extremely rare in a normally distributed data.