Miles per gallon. In its Fuel Economy Guide for 2008 model vehicles, the Environmental Protection Agency gives data on 1152 vehicles.
There are a number of outliers, mainly vehicles with very poor gas mileage.
If we ignore the outliers, however, the combined city and highway gas mileage of the other 1120 or so vehicles is approximately Normal with mean 18.7 miles per gallon (mpg) and standard deviation 4.3 mpg.
The quartiles of any distribution are the values with cumulative proportions 0.25 and 0.75.
They span the middle half of the distribution.
What are the quartiles of the distribution of gas mileage?
What is the first quartile of the distribution of gas mileage?
Use Table A and give your answer in mpg to 1 decimal place.
Ignoring the outliers, the quartiles are at ±.675 SD.
We do not have access to Table A.
To find the quartiles of the distribution of gas mileage, you can follow these steps:
Step 1: Calculate the z-scores
Z-score is a measure of how many standard deviations a data point is from the mean. Given a data point x, the z-score can be calculated using the formula:
z = (x - mean) / standard deviation
Step 2: Look up the cumulative probability in Table A
Table A provides the cumulative probabilities for the standard normal distribution. You can use this table to find the corresponding z-scores for the quartiles.
Step 3: Convert z-score to the actual value using the formula:
x = (z * standard deviation) + mean
Now let's calculate the first quartile of the distribution of gas mileage:
Step 1: Calculate the z-score for the first quartile.
The cumulative proportion for the first quartile is 0.25. Using Table A, you can find the z-score corresponding to a cumulative proportion of 0.25. Let's assume the first quartile z-score is z1.
Step 2: Convert z1 to the actual value using the formula.
Using the formula x = (z * standard deviation) + mean, substitute z1, standard deviation (4.3 mpg), and mean (18.7 mpg) to calculate the first quartile.
Step 3: Round the answer to 1 decimal place.
The answer should be rounded to the nearest tenth.
Using the given information and following the steps above, the first quartile of the distribution of gas mileage can be found as follows:
Step 1: Find z1 from Table A using a cumulative probability of 0.25.
From Table A, a cumulative probability of 0.25 corresponds to a z-score of approximately -0.674.
Step 2: Calculate the first quartile value (x1) using the formula.
x1 = (z1 * standard deviation) + mean
= (-0.674 * 4.3) + 18.7
≈ 15.24
Step 3: Round the answer to 1 decimal place.
The first quartile of the distribution of gas mileage is approximately 15.2 mpg.