Question

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. At what percentile rank is a car that sold for $30,000?

Answer 1

To find the percentile rank of a $30,000 car, the z-score is first calculated by dividing the difference between the car's price and the mean price by the standard deviation. With a z-score of 2, we then reference a z-table to determine the percentile, which is around the 97.5th percentile. This means that the car costs more than 97.5% of new cars.

Step-by-step explanation:

The student's question involves finding the percentile rank of a car that sold for $30,000 given that the average (mean) new car costs $23,000 with a standard deviation of $3,500, assuming that the prices are normally distributed. To solve this, we first calculate the z-score for $30,000 using the formula:

Z = (X - μ) / σ

Where X is the value in question ($30,000), μ is the mean ($23,000), and σ is the standard deviation ($3,500).

Z = ($30,000 - $23,000) / $3,500 = $7,000 / $3,500 = 2

Next, we refer to the standard normal distribution table (z-table) to find the percentile rank that corresponds to a z-score of 2. This z-score is typically associated with the 97.5th percentile (however, please verify with a standard normal distribution table as exact values can vary slightly).

Therefore, a car sold for $30,000 is at the 97.5th percentile rank, indicating that it costs more than 97.5% of new cars.

Answer 2

98th percentile.

Step-by-step explanation:

We have been given that an average new car costs $23,000, with a standard deviation of $3,500.

Since the price of cars are normally distributed, so we will find z-score for a car sold for $30,000 using z-score formula.

`z=(x-\mu)/(\sigma)`, where,

`z=\text{z-score}`,

`x=\text{Random sample score}`,

`\mu=\text{Mean}`,

`\sigma=\text{Standard deviation}`.

Substitute the given values:

`z=(\$30,000-\$23,000)/(\$3,500)`

`z=(\$7,000)/(\$3,500)`

`z=2`

Now, we will use normal distribution table to find what percent of data is below a z-score of 2.

`p(z<2)=0.97725`

Convert 0.97725 to percentage:

`.97725* 100\%=97.725\%\approx 97.73\%`

Therefore, the car that is sold for $30,000 has a percentile rank of 98.