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Mean Analysis

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Introduction

If we use sample mean (the mean of the sample) to approximate the population mean (the mean of the population), errors will be introduced.

Two questions:- How good is the approximation?- If the error is too large, what should we do

to reduce it?

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Confidence interval

Confidence interval gives us the interval in which the population mean would most likely fall.

The confidence level tells us (intuitively) the possibility the population mean will be in the confidence interval.

The smaller the confidence interval and the higher the confidence level, the better the approximation.

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Calculation of the confidence interval

The equation for confidence interval is:

The half-width of the C.I.: The point estimate is:

y z sn

z sn

y

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Some z values

Confidence level z

90% 1.65

95% 2

99% 2.6

99.7% 3

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How to find z value

Given a confidence level, say 80%, we need to find the Area value from the normal table that is closest to 80% and then find the corresponding z value, which will be the z value for the confidence interval formula.

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Properties

The larger the sample size n, the narrower the confidence interval.

The higher the confidence level, the larger z and thus the wider the confidence interval.

The population size has nothing to do with the confidence interval, as long as it is large enough.

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Distribution of the sample mean

If the sample size is large enough, say larger than 30, the histogram of the sample means is very close to a Normal distribution.

The mean of the sample means equals to the population mean.

The standard deviation of the sample means equals to the population standard deviation divided by the squared root of n.

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Conditions for Confidence Interval

The sampling method should be unbiased The sample size n should be large enough,

say larger than 30. The population size N should be much larger

than the sample size n.

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Determine the correct sample size n

1. Take a random sample of size n. 2. Calculate the confidence interval based on your

sample. 3. Check if the confidence interval is small enough. If

it is too wide, decide a proper width. 4. Assuming the sample standard deviation will

remain the same, use the equation for confidence interval to estimate the required sample size.

5. Take a new sample and calculate the confidence interval.

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When n is small

When n is small, say, n=5, a formula for the confidence interval can be derived only when the population distribution is normal.

The formula for confidence interval is very similar to the one we have used, the difference is we will have to employ the so-called Student t distribution instead of normal distribution.

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The Confidence interval for population proportion

The equation is:

– n is the sample size, – z is determined by the confidence level, e.g., if the

confidence level is 95% z is equal to 2, and

– p is the proportion of yes answers in the sample.

p z p p

n ( )1

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Conditions for Confidence Interval

The sampling method should be unbiased The sample size n should be large enough. In

particular, both np and n(1-p) should be larger than 5.

The population size N should be much larger than the sample size n.

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Comparing the means of two populations

The formula is:

– y2 and y1 are the sample means;

– s1 and s2 are the standard deviations of the two samples;

– n1 and n2 are the sample sizes of the two samples; and

– z is selected to provide the desired confidence level.

( )y y zs

n

s

n2 122

2

12

1

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Conditions for Confidence Interval

The two samples should be independently and randomly selected;

The sample sizes should be large enough – at least 30;

The population sizes should be much larger than the corresponding sample sizes.

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Compare proportions of two means

The formula is the following:

( )( ) ( )

p p zp p

n

p p

n2 12

12

2

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1

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