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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?

User Villi
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2 Answers

4 votes

Final answer:

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.

User Ralien
by
8.3k points
1 vote

Answer:

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.

User Marc Claesen
by
8.3k points

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