Using empirical rule and sample numbers find

Question

A car dealer notices that 10% of the cars he sells are metallic grey. Twenty of his customers are selected at random.

In a sample of 100 customer records find;

a) the number of expected metallic grey cars

b) the standard deviation of metallic grey cars

According to the empirical rule 95% of the orders of metallic grey are between a and b.

c) find a and b

Answer 1

I'm a little confused by this question, I think there may be a mistake. It says "20 of his customers are selected at random," but it also says "In a sample of 100 customer records." So which is it? Is the sample size 20 or 100? I'll proceed as if the sample size is 100, but you can adjust accordingly.

a) The number of metallic grey cars follows a binomial distribution with n = 100 and p = 0.1. In general, the mean of a binomial random variable is n*p, so in our case the mean is 100*0.1 = 10.

b) The standard deviation of a binomial random variable is √(n*p*(1-p)), in our case that's √(100*0.1*(1-0.1)) = 3.

c) According the empirical rule, about 95% of observations will be within 2 standard deviations from the mean. In other words, the lower bound for 95% of observations should be around (mean)-2*(standard deviation), and the upper bound should be (mean)+2*(standard deviation). If we plug in the numbers we found in part a and part b, we find that 95% of observations should be between 10-2*3 and 10+2*3, or 4 and 16.