Question

The Ford prices at a car dealership are normally distributed with a mean of $25,000 in a standard deviation of $1000. The BMW prices at the dealership are normally distributed with a mean of $70,000 and a standard deviation of $3000. Mr. X bought a Ford for $24,000 and Miss Y bought a BMW for $65,000. Find the z score for each and compare. Who got a better deal

Answer 1

We have to find which got a better deal.

We consider it a better deal when it has a lower value in respect to the possible prices of that specific brand.

We can compare the relative position of each deal by calculating the z-score. This z-score will give us the distance, as a proportion of the standard deviation, of the deal to the mean of the population.

Mr. X bought a Ford for $24,000. The mean and standard deviation of Ford prices is $25,000 and $1,000.

Then, we can calculate the z-value as:

`z=(X-\mu)/(\sigma)=(24000-25000)/(1000)=(-1000)/(1000)=-1`

We can visualize this as: there is 15.87% chances of getting a better deal with a Ford, as 15.87% of the prices are below $24,000.

NOTE: the z-score of -1 in this case means that $24,000 is one standard deviation to the left from the mean. The z-score gives us the relative position of a data point in a normal distrubution, letting us use a standard normal distribution to calculate the probabilities.

Miss Y bought a BMW for $65,000. The prices of BMW are normally distributed with a mean of $70,000 and a standard deviation of $3,000.

Then, we can calculate the z-score for this deal as:

`z=(X-\mu)/(\sigma)=(65000-70000)/(3000)=(-5000)/(3000)=-1.67`

The value of the z-score is lower than the one from Mr. X. We can see that in this case, the proportions of BMW's that cost less than $65,000 is 4.75%.

Then, the deal of Miss Y is better that the deal done by Mr X: there are less better deals for a BMW for Miss Y with the proce of $65,000 than the deals for less than $24,000 for a Ford.

Answer:

Z-score for Mr. X: -1

Z-score for Miss Y: -1.67

Miss Y got a better deal.