If the monthly salary of 1000 families is a normal random variable with mean 1700 Egyptian pounds and standard deviation of 200 Egyptian pounds, how many of those families have salaries of more than 1500 Egyptian pounds?
We can express 𝑥, a normal random variable, in terms of its mean and its variance. Its variance is its standard deviation squared. For us, 𝑥 equals the monthly salaries. 𝑥 follows a normal distribution with a mean of 1700 and a variance of 200 squared. Before we find the number of families who have a salary of more than 1500 pounds, we need to calculate the probability that a randomly chosen family will have a salary of larger than 1500 pounds. Because we know that 𝑥 follows a normal distribution, it follows a bell-shaped curve that is symmetrical about its mean.
We know that the mean is 1700. We know that the area under the full curve is one. And we’re interested in how many families earn more than 1500 pounds a month. This would be the area to the right of 1500. The area shaded in pink is the probability that 𝑥 is greater than 1500. To find this probability, we first need to know a 𝑧-score for the value we’re interested in. The 𝑧-score tells us how many standard deviations a particular value is above or below the mean. We find it by taking the value of 𝑥, subtracting the mean, and then dividing that by the standard deviation.
In this question, the value we’re interested in is 1500. So we’ll subtract the mean, 1700, from 1500 and then divide by 200. We get negative 200 over 200, which is negative one. This means that 1500 is one standard deviation to the left of the mean. And this make sense to us because 1500 is 200 away from 1700. From here, we’ll need to turn to a table of areas. Our table of areas looks something like this. This table of areas is showing us the probability that some value falls between zero and a specified 𝑧-score. It’s considering a scenario like this. And that means that our table of areas doesn’t give us the score for negative one. But we know that this curve is symmetrical. And that means the area under the curve from zero to one is equal to the area under the curve from zero to negative one.
So we look at positive one on the chart and we get 0.3413. The area of the space under the curve from 1500 to 1700 is 0.3413. And we know that the area to the right of the mean is one-half. And that means the probability that a randomly chosen family’s salary is greater than 1500 would be 0.3413 plus half. When we add those together, we get 0.8413.
Remember that our question is asking how many of those families would have salaries of more than 1500 Egyptian pounds. To find this, we take the 1000 families, and we multiply that by the probability that a randomly chosen family would have a salary greater than 1500. So we multiply 1000 by 0.8413 which gives us 841.3. Our number of families have to be an integer. We need to round this to the nearest whole number. 841.3 rounds down to 841. We can say the 841 families had a salary of more than 1500 Egyptian pounds.