Studies shown that gasoline use or compact cars sold in the U.S. is normally distributed, with a mean of 25.5 mpg and standard deviation of 4.5 mpg. Find the range of mileage for the middle 60% of compact cars.
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability (.30) from the mean and its Z score. Enter ±Z and other data into the equation above to get the scores.
To find the range of mileage for the middle 60% of compact cars, you need to determine the values that represent the lower and upper boundaries of this range.
First, let's find the Z-scores for the lower and upper quartiles of the distribution.
The middle 60% of the distribution corresponds to 60% / 2 = 30% on each side.
Step 1: Finding the Z-score for the lower quartile:
To find the Z-score for the lower quartile, you need to calculate the cumulative probability up to that point.
The cumulative probability represents the proportion of values that lie below a certain point in a normal distribution.
In this case, we want to find the lower boundary for the range, which is the Z-score that corresponds to the cumulative probability of 30%.
Using a standard normal distribution table or a calculator, you can find that the Z-score for a cumulative probability of 30% is approximately -0.524.
Step 2: Finding the actual mileage for the lower quartile:
Now that we have the Z-score, we can use it to find the actual mileage for the lower quartile.
The Z-score formula is: Z = (X - μ) / σ
Where:
Z = Z-score
X = Actual data value
μ = Mean of the distribution
σ = Standard deviation of the distribution
Rearranging the formula, we can solve for X:
X = Z * σ + μ
Plugging in the values we know:
X = -0.524 * 4.5 + 25.5
Calculating this expression, we find that the actual mileage for the lower quartile is approximately 23.39 mpg.
Step 3: Finding the Z-score for the upper quartile:
Similarly, we want to find the Z-score that corresponds to the cumulative probability of 70%, which is the upper quartile.
Using the standard normal distribution table or a calculator, we can find that the Z-score for a cumulative probability of 70% is approximately 0.524.
Step 4: Finding the actual mileage for the upper quartile:
Using the same formula as above, we can find the actual mileage for the upper quartile.
X = Z * σ + μ
Plugging in the values:
X = 0.524 * 4.5 + 25.5
Calculating this expression, we find that the actual mileage for the upper quartile is approximately 28.55 mpg.
Therefore, the range of mileage for the middle 60% of compact cars is from approximately 23.39 mpg to 28.55 mpg.