2. The results of a recent survey indicate that the average new car costs $23,000, with a standard deviation of $3,500. The price of cars is normally distributed.
a. What is a Z score for a car with a price of $ 33,000?
9.415000
b. What is a Z score for a car with a price of $30,000?
8.557857
c. At what percentile rank is a car that sold for $30,000?
Z = (score-mean)/SD = (33,000-23,000)/3500 = 10,000/3,500 = 2.86
Do the same for 30,000.
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion related to the second Z score.
To find the percentile rank of a car that sold for $30,000, we need to first calculate the Z-score for this price.
The Z-score formula is given by:
Z = (x - μ) / σ
Where:
Z is the Z-score
x is the value we want to calculate the Z-score for (in this case, $30,000)
μ is the mean of the distribution (average price of a new car, $23,000)
σ is the standard deviation of the distribution ($3,500)
Using the formula, we can calculate the Z-score for a car with a price of $30,000 as follows:
Z = (30,000 - 23,000) / 3,500
Z = 7,000 / 3,500
Z = 2
Now that we have the Z-score, we can find the percentile rank associated with it. The percentile rank represents the percentage of data points that fall below a certain value. To find the percentile rank, we will utilize a standard normal distribution table or a calculator.
Looking up a Z-score of 2 in a standard normal distribution table or using a calculator, we find that the percentile rank associated with a Z-score of 2 is approximately 97.72%.
Therefore, a car that sold for $30,000 would be at the approximately 97.72th percentile rank.
To find the percentile rank of a car that sold for $30,000, we need to calculate the area under the normal distribution curve to the left of that price.
Step 1: Calculate the Z-score for a car with a price of $30,000 using the formula:
Z = (x - μ) / σ
Where:
x = price of the car
μ = mean price of new cars ($23,000)
σ = standard deviation of new car prices ($3,500)
Z = (30,000 - 23,000) / 3,500
Z ≈ 2.000
Step 2: Find the percentile using the Z-table or a statistical calculator.
The percentile rank for a Z-score of 2 is approximately 97.7%.
Therefore, a car that sold for $30,000 is at the approximately 97.7th percentile rank.