Question

1. according to the truecar.com website, the asking prices for cars of a certain model are approximately normally distributed with mean $35,987 and standard deviation $607.50.

a. the factory invoice price for this model is $35,625. about what percent of cars have asking prices below the factory invoice price?
b. the manufacturers suggested retail price (msrp) for this car model is at the 98th percentile of the distribution of asking price. find the value of the msrp.

Explanation:

Step 1: Identify the distribution and parameters

The asking prices are normally distributed with mean \(\mu = 35987\) and standard deviation \(\sigma = 607.50\) (note: there was a typo in the handwritten work, the standard deviation is 607.50, not 5007.50). We need to find the percentage of cars with asking prices below \(\$35625\). First, calculate the z - score using the formula \(z=\frac{x - \mu}{\sigma}\), where \(x = 35625\), \(\mu=35987\), and \(\sigma = 607.50\).

\(z=\frac{35625 - 35987}{607.50}=\frac{- 362}{607.50}\approx - 0.596\approx - 0.60\)

Step 2: Find the area to the left of the z - score

Using the standard normal distribution table (z - table), the area to the left of \(z=- 0.60\) gives the proportion of data below that z - score. Looking up \(z = - 0.60\) in the z - table, the value is approximately \(0.2743\) or \(27.43\%\).

(For part b, to find the 98th percentile:

Step 1: Find the z - score for the 98th percentile

The 98th percentile means that \(98\%\) of the data is below this value. Looking up the z - score in the standard normal table (or using a calculator) for which the area to the left is \(0.98\), we find that \(z\approx2.05\) (since the area to the left of \(z = 2.05\) is approximately \(0.9798\), close to \(0.98\)).

Step 2: Use the z - score formula to find \(x\) (MSRP)

The z - score formula is \(z=\frac{x-\mu}{\sigma}\), we can rearrange it to \(x=\mu+z\sigma\). Substituting \(\mu = 35987\), \(z = 2.05\), and \(\sigma=607.50\):

\(x=35987+(2.05\times607.50)=35987 + 1245.375=37232.375\approx\$37232.38\)

Part (a) Answer:

The percentage of cars with asking prices below the factory invoice price is approximately \(27.43\%\) (or using more precise z - score calculation, if we use \(z\approx - 0.596\), the area to the left is \(P(Z < - 0.596)\approx0.275\) or \(27.5\%\)).

Part (b) Answer:

The MSRP (98th percentile) is approximately \(\$37232.38\)

(Note: The handwritten work had a mistake in the standard deviation value, using 5007.50 instead of 607.50, which led to incorrect results. The correct standard deviation from the problem statement is 607.50)

Answer:

s:
a. Approximately \(27.4\%\) (or \(27\%-28\%\) range)
b. Approximately \(\$37232.38\)