# PROBABILITY DISTRIBUTION (CONTINUATION)

PROPERTIES OF THE NORMAL DISTRIBUTION

1. It depends on the mean (u) and standard deviation
2. The shape is bell-shaped
3. The function is continuous, hence the range is from –ά to + ά
4. 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 .

The normal distribution has a complicated equation, but it can be shown in shaded area under the shape

1. Values within 1 sd of the mean

EVALUATION

Using the standard deviation normal distribution table. Find the area under

1. Pr (-1.5 < z< 2.0)     2.Pr(z  <1.5)              3.Pr (z >2.6)

Z Scores

The area under a normal distribution curve between two values depends on the number of standard deviations from the mean.  Therefore, the standardize normal curve is obtained from the normal curve by the substitution.

Z =   X – µ

σ

:.  Z is called the standardized score or Z score at mean zero (o) and standard deviation 1.

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

1. 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.
2. Use the standard normal distribution table to find  (i) Pr (Z > 2.6)    (ii) Pr (- 1.5 < Z < 1.7)

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

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