Statistics Exercises: Normal Distribution

Exercise 01: Call Center Call Time

Suppose that the time needed for customer service calls in a telephone call center follows a Normal distribution with a mean of 8 minutes and a standard deviation of 2 minutes.

1. What is the probability that a call lasts less than five minutes?
2. What is the probability that a call lasts more than 9.5 minutes?
3. What is the probability that a call lasts between 7 and 10 minutes?
4. 75% of calls require at least how long for service?

Exercise 02: Rabbit Weight Classification

The distribution of the weights of rabbits raised on a farm may well be represented by a Normal distribution with a mean of 5 kg and a standard deviation of 0.9 kg. An abattoir buys 5000 rabbits and intends to classify them according to weight as follows: 15% lighter as small, 50% as average, the next 20% as large, and 15% heavier as extras. What are the weight limits for each classification?

Exercise 03: Soft Drink Bottle Volume

An automatic filler for soft drinks is regulated so that the average volume of liquid in each bottle is 1000 cm³, and the standard deviation is 10 cm³. Assume the volume follows a Normal distribution.

1. What percentage of bottles have a liquid volume less than 990 cm³?
2. What percentage of bottles have a liquid volume that does not deviate from the mean by more than two standard deviations?
3. If 10 bottles are randomly selected, what is the probability that at most 4 have a fluid volume greater than 1002 cm³?
4. If bottles are being selected until one appears with a fluid volume greater than 1005 cm³, what is the probability of having to select at least 5 bottles?

Exercise 04: TV Defect Time and Profit

A company produces two types of televisions, type A (common) and type B (luxury), and guarantees a refund of the amount paid if any TV shows a serious defect within six months. The time to occurrence of a severe defect in TVs has a Normal distribution. For type A, the mean is 10 months and the standard deviation is 2 months. For type B, the mean is 11 months and the standard deviation is 3 months. Televisions type A and B are produced with a profit of 1200 and 2100 respectively. If there is a refund, there is a loss of 2500 for type A and 7000 for type B.

1. Calculate the probability of type A and type B TVs being returned for a refund.
2. Calculate the average profit for type A and type B televisions.
3. Based on average profits, should the company encourage sales of type A or type B appliances?