# The A to Z of mathematics a basic guide - Sidebotham T.H.

ISBN 0-471-15045-2

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68% of 320 = 0.68 x 320 Writing 68% as 0.68 = 217.6

jji -\-2a = 57.5 + 2 x 7.3 = 72.1 jji — 2cr = 57.5 — 2 x 7.3

42.9

-2o (x 2o (e)

NORMAL DISTRIBUTION 303

Useful Descriptors The normal distribution is applied to the study of probability. If a student is selected at random from William’s year group, there is a probability of 0.68 (68%) he or she will weigh between 50.2 and 64.8 kg. We say he or she will “probably” or “likely” weigh within one standard deviation of the mean. Similarly, there is a probability of 0.95 (95%) that a randomly selected student will weigh within two standard deviations of the mean. We say he or she will “very probably” or “very likely” lie within this range. There is a probability of 0.99 (99%) that a randomly selected student will weigh within three standard deviations of the mean. We say he or she will “almost certainly” lie within this range.

Example. The owner of the Choco Chocolate Company asked his foreman, John, to do an assurance check on the weights of their medium-size chocolate bars. John randomly selected 500 bars and after weighing them calculated that the sample had a mean weight of 147 grams and a standard deviation of 3 grams.

Assuming that the weights of Choco bars are normally distributed, answer the following:

(a) What is the weight of a bar of Choco very likely to be?

(b) If a bar is randomly selected from a shelf in a supermarket, what is the probability it will weigh more than 150 grams?

(c) The owner of the company decides to print the following guarantee on the wrapper of each Choco bar:

“This bar is guaranteed to weigh 141 grams, or your money back.”

How many bars in the sample of 500 will fail the guarantee?

Solution. The first step is to draw a normal curve, complete with all the known information regarding percentages and means and standard deviations (see figure g):

fx = 147, fx + cr = 150, jx — a — 144, fx + 2a = 153, jx — 2a = 141, jx + 3cr = 156, fx — 3a = 138

304

NORMAL DISTRIBUTION

(9)

(a) “Very likely” is within two standard deviations of the mean, which is between 141 and 153 grams. Abar of Choco is very likely to weigh between 141 and 153 grams.

(b) The probability that a bar will weigh more than 150 grams is 13.5% + 2% + 0.5% = 16%. The probability that a Choco bar will weigh more than 150 grams is 16%, which is 0.16.

(c) The percentage of bars that weigh less than 141 grams is 2% + 0.5% = 2.5%. The number in the sample of 500 bars that weigh less than 141 grams is equal to 2.5% of 500: 0.025 x 500 = 12.5. About 13 of 500 Choco bars will be expected to fail the guarantee.

The type of data that result in a normal distribution curve are continuous data, but the properties of a normal curve can sometimes be applied to discrete data. For example, the examination marks of a large number of students display the characteristics of a normal distribution.

The normal curve can be used in quality control, as explained in this next example. Peter is a baker and is concerned with the fluctuations in the weights of his standard loaf of bread. He employs Lisa as a quality control officer to improve the consistency in this area. On a Tuesday, Lisa selects a random sample of 1000 loaves and weighs them. Then she enters the data in a frequency table and draws a normal distribution curve. Lisa implements her quality control system in the bakery to ensure the weights of the standard loaves will be less variable. Three months later, on a Tuesday, she checks the weights of 1000 loaves and draws another normal distribution curve on the same axes as the first batch. It can be seen from the graphs in figure h that the second batch has weights that are less spread out and more closely clustered about the mean weight. Her quality control systems are working.

(h)

References: Arithmetic Mean, Axis of Symmetry, Central Tendency, Class Interval, Continuous

Data, Frequency Distribution, Frequency Polygon, Histogram, Standard Deviation.

NUMERICAL VALUE 305

NORTH

Reference: East.

NUMBER BASES

References: Binary Numbers.

NUMBER LINE

Reference: Directed Numbers, Integers.

NUMBER PAIRS

These are sometimes called ordered pairs.

Reference: Cartesian Coordinates.

NUMBER TREE

Reference: Integers.

NUMBERS

Reference: Integers.

NUMERATOR

Reference: Denominator.

NUMERICAL VALUE

The equation 3x + 2 = 8 is a numerical equation, because the constants are all numbers. The equation ax2 + bx + c = 0 is called a literal equation, because its constants are represented by the letters a, b, and c.

The term numerical value is sometimes used instead of absolute value. For example, the numerical value of —7 and +7 is 7.

References: Absolute Value, Equations.

o

OBJECT

Reference: Image.

OBTUSE ANGLE

Reference: Acute Angle.

OCTAGON

An octagon is a polygon with eight sides; the prefix octa means eight. The angle sum of the eight interior angles of any octagon is 1080°. Some octagons are drawn in figure a; the last one is a regular octagon.

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