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!
Well, if the dealer's service manager claimed that 17.2 mpg is normal for this model car, you could respond by saying "Well, that's one way to drive people away from buying cars!" But in all seriousness, you could refute his claim by using statistics. Since the actual mileage is normally distributed around an average of 22.7 mpg with a standard deviation of 1.4 mpg, you can calculate the z-score for a mileage of 17.2 mpg.
Using the formula z = (x - μ) / σ, where x is the observed value, μ is the mean, and σ is the standard deviation, you can plug in the values:
z = (17.2 - 22.7) / 1.4
Once you calculate the z-score, you can compare it to the standard normal distribution and see what percentage of values fall below that z-score. Based on that, you can conclude whether 17.2 mpg is a normal or abnormal value for this model car.
To answer question f, you can use z-scores and the concept of standard deviations from the mean to determine if a mileage of 17.2 mpg is statistically significant or not. Here's how you can approach it:
1. Calculate the z-score:
The z-score is a measure of how many standard deviations a particular value is away from the mean. It is calculated using the formula:
z = (x - μ) / σ
Where:
- x is the value you want to test (17.2 mpg in this case)
- μ is the mean (22.7 mpg in this case)
- σ is the standard deviation (1.4 mpg in this case)
Substituting the given values, we get:
z = (17.2 - 22.7) / 1.4
z ≈ -3.93
2. Determine the significance level:
The significance level is the threshold beyond which a value is considered statistically significant. Typically, a significance level of 0.05 (or 5%) is used.
3. Look up the critical z-value:
The critical z-value corresponds to the specific significance level chosen. For a significance level of 0.05, the critical z-value is approximately ±1.96 (from a standard normal distribution table).
4. Compare the z-score and critical z-value:
If the absolute value of the z-score is greater than the critical z-value, then the value is considered statistically significant. In this case, |z| = 3.93 > 1.96, indicating that the mileage of 17.2 mpg is statistically significant.
5. Refuting the claim:
Based on the statistical analysis, you can refute the dealer's claim that a mileage of 17.2 mpg is normal for this model of car. The low mileage is statistically significant, suggesting that it is outside the expected range.
To further support your argument, you can mention that the expected range for 80% of the mileage is between 20.9 and 24.5 mpg (as answered in question d). Since 17.2 mpg is well below this range, it provides additional evidence against the claim of normality.
Note: It's important to keep in mind that this analysis assumes that the distribution of mileage is normally distributed and that the data provided accurately reflects the population.