The table shows the number of cars that 65 families have. Find the mean number of cars per family.
To find the mean from a frequency table, we can apply the given formula. In this case, 𝑓 is the frequency and 𝑥 is the number of cars each family owns. Let’s begin by finding the sum of the products of the number of cars and their frequency.
For the first column, that’s one multiplied by 10. We then have 35 families who own two cars, so that’s two multiplied by 35; 15 families who own three cars — that’s three multiplied by 15; and for the final column, that’s four multiplied by five. That’s 10 plus 70 plus 45 plus 20, which is equal to 145.
Now we don’t actually need to work out the total frequency, but if we did, we could just add 10, 35, 15, and five. In fact, we’re told that there are 65 families, so we can simply divide 145 by 65. 145 divided by 65 is twenty-nine thirteenths.
We can check whether the answer is likely to be correct by considering the context of our question. Twenty-nine thirteenths is a little over two, and the number of cars per family were either one, two, three, or four. So twenty-nine thirteenths is a sensible answer for the mean number of cars per family.
The data can be expressed as a probability distribution for the discrete random variable 𝑋 as shown. Find the value of 𝑎, 𝑏, 𝑐, and 𝑑.
To answer this, we’ll need to consider the relative frequency for each value of 𝑋. When the number of cars per family is one, 𝑥 is one. There were 10 families who had one car and a total of 65 families in this study. This means the relative frequency or the probability that a person chosen at random from this group had one car is 10 over 65, which simplifies to two thirteenths, so 𝑎 is two thirteenths.
We can repeat this process to find the values of 𝑏, 𝑐, and 𝑑. 𝑏 is the probability that a family chosen at random has two cars. There were 35 families who had two cars, so the probability that a family chosen at random has two cars is 35 out of 65, which simplifies to seven thirteenths. There were 15 families who had three cars, so the probability a person chosen at random had three cars is 15 over 65. That means that 𝑐 is three thirteenths. And finally, 𝑑 represents the probability that a family chosen at random has four cars. There were five families who had four cars, so the probability of choosing a family at random with four cars is five over 65, which is one thirteenth. 𝑎 is two thirteenths, 𝑏 is seven thirteenths, 𝑐 is three thirteenths, and 𝑑 is one thirteenth.
Calculate the expected value of 𝑋.
To find the expected value of a discrete random variable 𝑋, we find the sum of the products of 𝑋 multiplied by the probability of 𝑋 occurring. In this case, that’s one multiplied by two thirteenths, two multiplied by seven thirteenths, three multiplied by three thirteenths, and four multiplied by one thirteenth, which is twenty-nine thirteenths.
Notice the expected value of 𝑋 is simply the mean, so we could’ve used the answer from the first part of this question to find the expected value of 𝑋. Either way, the expected value of 𝑋 is twenty-nine thirteenths.