Question

Pay your bills. In a large sample of 250 customer accounts, a

In a large sample of 250 customer accounts, a utility company determined that the average number of days between when the bill was sent out and when the payment was made is 32 with a standard deviation of 7 days. Assume the data to be approximately bell- shaped.

1. Use the empirical rule to construct a diagram of the distribution of the data.
2. Between what two numbers will approximately 68% of the numbers of days be?
3. Estimate the number of customers’ accounts for which the number of days is between 11 and 53.
4. Estimate the number of customers’ accounts that were greater than 39.
5. Estimate the number of customers’ accounts that were between 25 and 53.

Answer 1

a.

b.

Approximately 68% of the numbers of days will be between and i.e. 32-7 = 25 and 32+7 = 39

c.

z = (11-32) / 7 = -3

z = (53-32) / 7 = 3

The number 11 and 53 are 3 standard deviations away from mean. Thus, by empirical rule, approximately 99.7% of the number of customers’ accounts for which the number of days is between 11 and 53.

Number of customers’ accounts for which the number of days is between 11 and 53 = 0.9957 * 250 = 249

d.

z = (39 - 32) / 7 = 1

By empirical rule and symmetry of the bell distribution, the proportion of number of customers’ accounts that were greater than 39 = (1 - 0.68)/2 = 0.16

Number of customers’ accounts that were greater than 39 = 0.16 * 250 = 40

e,

z = (25 - 32)/7 = -1

z = (53-32)/7 = 3

By empirical rule and symmetry of the bell distribution, the proportion of number of customers’ accounts that were between 25 and 53 = 0.9957/2 + 0.68/2 = 0.83785

Number of customers’ accounts that were between 25 and 53 = 0.83785 * 250 = 209