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?
b. What is a Z score for a car with a price of $30,000?
c. At what percentile rank is a car that sold for $30,000?
a and b. Z = (score-mean)/SD
c. Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion related to the Z score.
To find the Z score for a given value in a normal distribution, we can use the formula:
Z = (X - μ) / σ
Where:
- Z is the Z score
- X is the value we want to find the Z score for
- μ is the mean of the distribution
- σ is the standard deviation of the distribution
a. To find the Z score for a car with a price of $33,000, we can use the given information:
- Mean (μ) = $23,000
- Standard deviation (σ) = $3,500
- Value (X) = $33,000
Plugging these values into the formula, we have:
Z = ($33,000 - $23,000) / $3,500
Z = $10,000 / $3,500
Z ≈ 2.857
Therefore, the Z score for a car with a price of $33,000 is approximately 2.857.
b. To find the Z score for a car with a price of $30,000, we can use the same formula:
- Mean (μ) = $23,000
- Standard deviation (σ) = $3,500
- Value (X) = $30,000
Plugging these values into the formula, we have:
Z = ($30,000 - $23,000) / $3,500
Z = $7,000 / $3,500
Z = 2.000
Therefore, the Z score for a car with a price of $30,000 is 2.000.
c. To find at what percentile rank a car that sold for $30,000 is, we can use the Z score. The Z table can provide us with the percentile rank corresponding to a given Z score. Looking up the Z score of 2.000 in the Z table, we find that it corresponds to a percentile rank of approximately 97.72%.
Therefore, the car that sold for $30,000 is at the approximately 97.72th percentile rank.