Sampling Sampling distributions

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QMS 204: STATISTICS FOR MANAGERS l Instructor: Moez Hababou

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Statistical Inference l

The use of random samples from a population to make inferences about a population

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Statistical Inference l l l

Two types of inference procedures Estimation Hypothesis Testing

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Random Sample l

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All individuals or objects in the population have a chance of being selected The selection of any one individual or object does not affect the selection probability for any other individual or object

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Simple Random Sample l

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A random sample selected in such a way that all possible samples of the same size have the same chance of being selected Implies that all individuals have the same chance of being selected for the sample

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Finite Populations and Sampling With Replacement l

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We assume that samples are drawn without replacement and that the population is large enough that the number of possible samples of size n is still large If the sampling fraction n/N ( where N is the population size) is less than 5% then we ignore the finite population effect

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Finite Populations and Sampling With Replacement l

If the population is small and finite corrections have to be made(not to be discussed in this course)

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Population Distribution l

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This is the distribution of all items in the population of interest It is characterized by the values of certain parameters which are usually unknown

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Examples Population Distribution X the random variable

µ population mean σ2 population variance draw samples

Sample 1 of size n

µ σ

X

Sample 2 of size n

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Sample Distribution l

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This is the distribution of the items that appeared in the sample This information is used to derive the values of sample statistics which are used to estimate the unknown parameters

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Examples Population Distribution X the random variable µ population mean

σ2 population variance Sample of size n x1, x2, x3 …., x n ,

µ σ

X

Sample distribution

x

x s

S2

X

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Sampling Distribution l

This is the distribution of a sample statistic over all possible samples of the same size taken from the same population

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Examples Sampling distribution of

x

µ

Sampling distribution of S 2

x

σ2

S2

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Statistical Inference For The Population Mean l

Requires the sampling distribution of the mean

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Sampling Distribution of The Sample Mean l

This is the distribution of the sample mean over all possible simple random samples of the same size taken from the same population

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Normal Population Variance Known l

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The sampling distribution of the mean is a Normal distribution with mean µ and standard deviation σ/√n The standard deviation of the mean σ/√n is called the standard error of the mean As the sample size n increases the variation of the sample means around the true mean decreases page16

Examples Distribution of x Small Sample • more variation in

Large Sample • less variation in

x

µ

x

σ/ n

x

µ

x

σ/ n page17

Examples l

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It has been determined that heating expenses are normally distributed with a mean of $500.00 and a standard deviation of $125.00. a) What is the probability that in a random sample of 25 accounts the average balance could exceed $550?

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Examples

Z = 550 - 500 = 2

125/ 25

500

550

P[ Z > 2 ] = 0.0228

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Examples l

b) What is the probability that in a random sample of 25 accounts the average would lie between $475.00 and $540.00? Z = 540 - 500 = 1.60

125/ 25 Z = 475 - 500 = -1.00 475

500

540

125/ 25

P[ -1 ≤ Z ≤ 1.60 ] = 0.7865 page20

Nonnormal Population l

Central Limit Theorem

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Central Limit Theorem l

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In large samples the central limit theorem states that the sample mean is normally distributed with mean µ and standard deviation σ/√n Thus in large samples we need not be concerned about whether the underlying population is normal

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Examples l

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A random sample of 64 residents of a particular section of the city were asked to indicate the number of hours per week they watch television. The population standard deviation is 16. a) What is the probability that the sample mean will exceed the population mean by 2 in absolute value? page23

Examples

µ- 2 µ µ+2

Z=

(µ+2) -

µ

= 1

Z=

16 / 64 (µ-2) - µ

= -1

16 / 64

P[ |Z| > 1 ] = 2(0.1587) = 0.3174 page24

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Examples l

b) If the sample size were 144 what would be the probability in a)?

µ- 2

µ

µ+2

Z=

(µ+2) -

Z=

(µ-2) -

µ

16/ 144

µ

16/ 144

= 1.5 = -1.5

P[ |Z| > 1.5 ] = 2(0.0668) = 0.1336 page25

Sampling Distribution for the Mean When Variance Unknown l

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Population variance unknown in large samples Population variance unknown in small samples

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Population Variance Unknown in Large Samples l

By the central limit theorem in large samples the statistic below has a standard normal distribution

x − µ s/ n l

This distribution is exact if the population being sampled is normal page27

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Population Variance Unknown in Small Samples l

In small samples if the population variance is unknown and the population is normal the statistic below has a t distribution with (n-1) degrees of freedom

x − µ s/ n

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If the population is not normal we do not know the distribution of this statistic page28

t-Distribution l

the t distribution is bell shaped and symmetrical and looks very much like a standard normal distribution Normal distribution

t distribution

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t-Distribution l

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the mean of the distribution is zero and the variance is larger than the standard normal variance of 1 as the number of degrees of freedom increase, the variance of the t distribution approaches 1

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t-Distribution l

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the critical values of the t distribution are larger than the corresponding critical values for the standard normal but approach the standard normal values as the number of degrees of freedom rise by the time d.f. = (n-1) = 30 the t distribution and standard normal are very close examine the critical values of t for α = 0.025 as the d.f. change page31

Sampling For Proportions l

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In the special case of a binomial population we are interested in making inferences about the binomial parameter π This is a special case of the central limit theorem In large samples the sampling distribution of the sample proportion p has a mean of π and a standard deviation of √π (1-π)/n page32

Sampling For Proportions l

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The sample proportion p is equivalent to a sample mean for a variable whose underlying values are either 0 or 1 In the case of a sample proportion the sample size needs to be large if the true proportion π is close to zero or 1 A useful rule of thumb is that both n π and n(1-π) should exceed 5 page33

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Examples If π = 0.10 n π > 5 ⇒ n > 50

If

π = 0.90 n(1- π) > 5 ⇒ n > 50

If π = 0.01 n π > 5 ⇒ n > 500

If

π = 0.99 n(1- π) > 5 ⇒ n > 500

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Examples l

Suppose that 60% of the population of Alberta voters wish to eliminate video lottery terminals. Suppose a sample of 100 voters was selected and asked for their opinion. What is the probability that less than 50% of the sample will be in favor of eliminating the terminals?

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Examples

Z = 0.50 - 0.60

= -2.04

(0 .60)(0 .40) 100 0.5

0.6

P[ Z < -2.04 ] = 0.0207

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