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OBJECTIVES

• To understand concept of sampling distribution

• To understand concept of sampling error

• To determine the mean and std dev for the

sampling distribution of a sample mean

• To determine the mean and std dev for sampling

distribution of a sample proportion

• To calculate the probabilities related to the

sample mean and the sample proportion

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Sampling distributions

• Can be defined as the distribution of a sample

statistic.

• Scientific experiments are used to make

inferences concerning population parameters from

sample statistics.

• Need to know what is the relationship between the

sample statistic and its corresponding population

parameter.

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Sampling error

• Can be defined as the difference between the

calculated sample statistic and population

parameter.

• Sampling errors occur because only some of the

observations from the population are contained in

the sample.

• Sampling error:

sample statistic – population parameter

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Sampling error

• Size of the sampling error depends on the sample

selected.

• May be positive or negative.

• Should be kept as small as possible.

• For smaller samples the range of possible

sampling errors becomes larger.

• For larger samples the range of possible sampling

errors becomes smaller.

CONCEPT QUESTIONS

• P201 QUESTIONS 1-4

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Sampling distribution of the mean

• Sample mean is often used to estimate the

population mean.

• Sampling distribution of the mean is the

distribution of sample means obtained if all

possible samples of the same size are selected

form the population.

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Sampling distribution of the mean

• If we calculate the average of all the sample

means, say we have m such samples, the result will

be the population mean:

• The standard deviation of all the sample means, will

be: referred to as the standard error of the mean

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m

i

ix

x

m

xn

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Central Limit Theorem

• If the sample size becomes larger, regardless of the

distribution of the population from which the sample

was taken, the distribution of the sample mean is

approximately normally distributed:

– with

– and standard deviation

• The accuracy of this approximation increases as

the size of the sample increases.

• A sample of at least 30 is considered large enough

for the normal approximation to be applied.

x x

n

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Properties of the sampling distribution of

the sample mean

• For a random sample of size n from a population

with mean μ and standard deviation σ, the

sampling distribution of has:

– a mean

– and a standard deviation

x

xn

x

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Properties of the sampling distribution of

the sample mean

• If the population has a normal distribution, the

sampling distribution of will be normally

distributed, regardless the sample size.

• If the population distribution is not normal, the

sampling distribution of will be approximately

normally distributed, if the sample size ≥ 30.

•

x

x

2

;X Nn

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Example

• Marks for a semester test is normally distributed, with a mean of 60 and a standard deviation of 8. – X ~ N(60;82)

• A sample of 25 students is randomly selected:

– 2

2 2

;

8; 60;

25

x xX N

X N Nn

Example • If we need to determine the probability that the

average mark for the 25 students will be between 58 and 63.

(58 63)

58 60 63 60

8 8

25 25

1,25 1,88

0,9699 0,8944 1 0,8643

P X

P Z

P Z

INDIVIDUAL EXERCISE

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INDIVIDUAL EXERCISE

The past sales record for ice cream indicates the sales are right skewed, with the population mean of R13.50 per customer and a std dev of R6.50. A random sample of 100 sales records is selected. Find the probability of:-

1. Getting a mean of less than R13.25

2. Getting a mean of greater than R14.50

3. Getting a mean of between R13.80 and R15.20

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Solution

P205 - 207 of textbook

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WHICH EQUATION TO USE?

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Sampling distribution of

proportion • Categorical values such as number of

drivers that wear safety belts in Gauteng

or number of drivers who do not wear

safety belts

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Sampling distribution of the proportion

• Population proportion will be represented by p,

and the sample proportion by , where X is

the number of items with the characteristic and n

is the sample size.

• The standard error of the proportion is given as:

ˆ /p X n

(1 )p

p p

n

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Example

• Suppose that in a class of 100, 28 students fail a test.

• The population proportion of students who fail the test is:

28

ˆ 0,28100

Xp

n

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Example

• A sample of 50 students is randomly chosen

• What is the probability that more than 25% will fail the test?

ˆ 0,25

0,25 0,28

0,28(1 0,28)

50

0,47

1 0,6808 0,3192

P P

P Z

P Z

Individual exercise/homework

• Read pages 195 – 211

• Self review test p 209

• Supplementary exercises p209

• Go to www.jillmitchell.net and view the following:-

• Video on sampling distributions

• Video on example of sampling distribution

• Video on central limit theorem

• Completely re-do the NUBE test using your textbook to

assist you.

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