Lecture 21

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Example: Election poll results expressed as 42 +/- 5 is an example of an interval estimate of a population mean

Interval estimate is better than a point estimate since it gives an idea of the precision of the estimate

 

What factors affect the width of the interval estimate?

 

Construction of the Interval Estimate

The interval estimate is written as

 

 

Where

The width of the interval consist of two parts

confidence value - k

standard deviation of X-bar

 

The confidence value (k) is SELECTED by the designer of the experiment and represents the degree of confidence we have that the interval will be correct.

Example: If the sampling distribution is normally distributed and we want a 95% confidence that the interval is correct then k = 1.96

A 95% CI means that 95% of intervals constructed for a given sample size will include the population mean. In other words 95% of the intervals will be correct

In practice, we have only one interval - in constructing our interval we want to ensure that the chance that it is correct is fairly highly

The standard deviation of X-bar is determined from the theorem for the variance of the sampling distribution.

 

But we don’t know the value of the population standard deviation (s) therefore we must use the sample standard deviation (s) as an estimate

Therefore, we use

 

 

 

 

The confidence value (k) depends on the actual distribution type for the normalized parameter

 

 

 

 

For Large Samples (n>100)

The distribution of the normalized parameter is normal

Therefore, k = z(1-a/2)

Where alpha is the level of significance (for example, 10% level of significance)

Why "alpha over two"?

Lecture 22

 

 

 

 

For Small Samples

If the population distribution is normal

The distribution (for the normalized parameter) is a t-distribution

This distribution is not normal because s is used to estimate s

For small samples s is not a reliable estimator of s

 

Therefore, k = t(1-a/2, n-1)

 

If the population distribution is not normal

The distribution (for the normalized parameter) is approximately a t-distribution

Therefore, k = t(1-a/2, n-1)

Therefore, k is approximately t(1-a/2,n-1)

 

Constructing Confidence Interval

 

In the design of an experiment we must determine the size of the sample needed to give us the desired degree of precision

The sample size depends on

the required precision (+/- h)

the level of confidence (90%, 95% etc)

the population standard deviation

How do we get the population standard deviation?

A guestimate based on (in best case) prior experience

The equation relating sample size to there parameters is n = (z²s²)/h²

Derive this equation!

 

 

 

Lecture 23

Confidence Interval for Population Proportions

The results of a poll is given as a proportion or percentage

The purpose of polling is to estimate the population proportion using a sample

The polling result is frequently given as confidence interval

The procedure for constructing the confidence interval is exactly the same as for population mean

The difference is that the sampling distribution of the population proportion is different from that of the population mean

 

Sampling Distribution of Population Proportion

Polling results from a binomial experiment

However, we are not interested in x (the number of successes) but in p (proportion of successes)

In other words, p = x/n

What is the population mean and variance of x for a binomial experiment?

mx = np

sx = sqrt (npq) (where q=1-p)

What is the population mean and variance for p?

mp = p

sp = sx / sqrt(n) = sqrt (pq)

 

Sampling Distribution of Population Proportion

The population proportion is p

The sample proportions is written as

Parameters of Sampling Distrbution of Sample Proportion

Mean: p

Standard Deviation: sqrt (pq/n)

Distribution Type: Normal for large sample size

When sample size is large - p-hat  is used to approximate p in calculating the standard deviation

Sample size is considered large if the interval

does not include 0 or 1

 

Confidence Interval of Population Proportion

For large sample CI is given by