- Normal distribution
- properties and area
- z – scores application.

**PROPERTIES OF THE NORMAL DISTRIBUTION**

- It depends on the mean (u) and standard deviation
- The shape is bell-shaped
- The function is continuous, hence the range is from –ά to + ά
- The curve is symmetrical about the vertical line through the mean.

A normal distribution function is a probability function, hence the total area under the curve is 1 . T

**EVALUATION**

The weights of packets of sugar produced by a machine have a mean of 1kg and a standard deviation of 0.1kg. What is the probability that in a random sample of 50 packets the combined weight will exceed 52kg?

**GENERAL EVALUATION**

- The inner diameters of bolts produced in a factory are normally distributed with mean 5cm and standard deviation 0.02cm. Find (a) the percentage of the number of bolts with inner diameters less than 5.015cm; (b) the probability that a bolt will have an inner diameter between 4.995cm and 5.015cm.
- Use the standard normal distribution table to find (i) Pr (Z > 2.6) (ii) Pr (- 1.5 < Z < 1.7)

READING ASSIGNMENT

Read Z scores and Normal distribution. Further Mathematics Project III Page 2 202-210.

**WEEKEND ASSIGNMENT**

1. Find the area between z = 0.36 and z= 1.89 (a) 0.33 (b) 0.6112 (c) 1.00

Use the information below to answer questions 2-4.

A distribution with mean 85 and standard deviation 10 is normally distributed. If x is a random variable of the distribution, find

2. Pr (80 < x <8.9) (a) 0.9332 (b) 0.5 (c) 0.3469

3. Pr(x >83) (a) 0.1587 (b) 0.789 (c) 0.4207

4. Pr(x > 87) (a) 0.0047 (b) 0.35 (c) 0.4207

5. Find, with the usual notations, P (z <1.810) from the table of normal distribution.

(a) 0.311 (b) 0.0288 (c) 0.9649

**THEORY**

- The scores of some 500 candidates in an examination were found to be approximately normally distributed with mean 40 and standard deviation 5. Find the number of candidates who scored at least 48.
- The lengths of nails produced in a factory are approximately normally distributed with mean 2cm and a standard deviation 0.01cm. Find the proportion of nails that will be shorter than 1.98cm.