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. I have no idea how to do this with the percent.
To find the range of mileage for the middle 60% of compact cars, we need to use the concept of a standard normal distribution. Specifically, we will use a z-table or a statistical calculator to find the z-scores corresponding to the given percentage.
Here's how you can calculate the range:
1. Find the z-scores for the lower and upper percentiles:
To find the z-scores, we need to find the cutoff points for the lower 20% and upper 20% of the distribution. Since we want the middle 60%, the remaining 100% - 60% = 40% will be split equally on both sides.
The lower percentile would be (100% - 40%)/2 = 30%, and the upper percentile would also be 30%.
Using a z-table or a statistical calculator, we find that the z-score corresponding to a cumulative probability of 0.30 is approximately -0.52 (for the lower percentile) and 0.52 (for the upper percentile).
2. Convert the z-scores to actual mileage values:
Since we have the mean and standard deviation, we can use the formula z = (x - μ) / σ, where z is the z-score, x is the value we want to find, μ is the mean, and σ is the standard deviation.
Rearranging the formula to solve for x, we get x = μ + (z * σ).
Plugging in the values, x1 = 25.5 + (-0.52 * 4.5) ≈ 23.86 and x2 = 25.5 + (0.52 * 4.5) ≈ 27.14.
3. Calculate the range of mileage for the middle 60%:
The range would be the difference between the upper and lower values, x2 - x1, which gives us 27.14 - 23.86 ≈ 3.28 miles per gallon.
Therefore, the range of mileage for the middle 60% of compact cars is approximately 3.28 miles per gallon.
To find the range of mileage for the middle 60% of compact cars, we need to calculate the z-scores corresponding to the lower and upper bounds of this 60% range.
Step 1: Convert the given 60% range into a 30% range on each side. Since the normal distribution is symmetric, the lower 30% ranges from the left tail to the lower bound of the middle 60%, and the upper 30% ranges from the upper bound of the middle 60% to the right tail.
Step 2: Convert the lower and upper percentiles to z-scores using the standard normal distribution table or calculator.
Step 3: Use the z-scores and the given mean and standard deviation to find the corresponding mileage values.
Let's calculate this step-by-step:
Step 1:
The middle 60% range is divided into two equal parts, so each side will have 30% of the data.
Step 2:
To find the z-scores corresponding to the lower and upper bounds of the 30% range, we need to find the z-values that correspond to the cumulative probability of 0.15 and 0.85.
Using a standard normal distribution table, the z-score corresponding to a cumulative probability of 0.15 is approximately -1.0364 (rounded to four decimal places). Similarly, the z-score corresponding to a cumulative probability of 0.85 is approximately 1.0364.
Step 3:
To find the mileage values corresponding to these z-scores, we can use the formula:
Mileage = Mean + (Z-Score * Standard Deviation)
For the lower bound of the middle 60% range:
Mileage_lower = 25.5 + (-1.0364 * 4.5)
= 25.5 - 4.664
= 20.839 (rounded)
For the upper bound of the middle 60% range:
Mileage_upper = 25.5 + (1.0364 * 4.5)
= 25.5 + 4.664
= 30.164 (rounded)
Therefore, the range of mileage for the middle 60% of compact cars is approximately 20.839 to 30.164 mpg.