## Normal Distribution Exercise

With all of these problems it is helpful to draw a diagram.

1. Use the standard normal tables to find the following probabilities.

a. P( 0 < Z < 1.3)

b. P( -1.42 < Z < 0)

c. P( -0.934 < Z < 1.528)

d. P( Z < 1.87)

2. A normal distribution has a mean μ = 16 and a standard deviation σ = 3. Use the formula  to help find the probabilities that:

a. P( 12 < x < 18.5)

b. P( x > 13)

c. P( x < 20)

 3. The weights of eggs from a certain farm are normally distributed with a mean weight of 73 g and a standard deviation of 5 g. If an egg which weighs over 70 g is classed as "large", what proportion of eggs from this farm are large?

4. A normal distribution of a continuous variable has a mean of 20.8 and a standard deviation of 2.9.

Find the probability that a value selected at random will between 16 and 24.

5. The time taken by a postman to post mail is normally distributed with a mean of 15 minutes and a standard deviation of 3 minutes. He delivers the post every day.

a. What is the probability that he takes longer than 18 minutes?

b. Assuming that he works 365 days a year, on how many of these days would you expect him to work between 14 and 22 minutes?

 6. The weights of kiwifruit from an export order to the US is normally distributed with a mean of 560 g and a standard deviation of 62 g. a. What percentage of all kiwifruit would be expected to weigh between 480 g and 640 g? b. In a load of 5000 kiwifruit, how many kiwifruit would be below 450 kg?

7. The loaded weight of a tour party's baggage was found to be normally distributed with a mean of 12.8 kg and a standard deviation of 2.5 kg. The optimum weight of a tourist's bag was said to be between 11 kg and 15 kg. What percentage of bags come in this range?

8. Tennis balls are tested by dropping them from a certain fixed height and measuring the height of the bounce. A ball is classed as satisfactory if it rises above 55 cm. The heights that the balls rise is normally distributed and the mean height of the bounce is 68 cm with a standard deviation of 7 cm. What percentage of balls will "fail" the test?

9. If a normal distribution of a continuous variable has a mean of 19.5 and a standard deviation of 2.95. Find the probability that a value selected at random will be larger than 25 or less than 14.

10. If the life of a certain make of battery is normally distributed with a mean life of 40 months and a standard deviation of 6 months, what percentage of these batteries can be expected to last between 25 to 35 months?