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bstatsmodule6slides.pptxModule 6 point estimators and sampling distributions
Master for Business Statistics
Dane McGuckian
Topics
6.1 Point Estimators and Sampling Distributions
6.2 The Central Limit Theorem
6.1
Point Estimators and Sampling Distributions
Sampling Distributions
The sampling distribution of a statistic is a probability distribution for all of the possible values of a sample statistic that can be derived from samples of a given size.
Recall that a probability distribution provides all possible outcomes for an experiment and the probability associated with each of these outcomes.
Example: If we took every possible random sample of 25 values from a population and calculated the sample mean for each sample, the resulting sampling distribution for the sample mean would provide all possible means that could result from a sample of 25 values drawn from this population along with the probability that each of those means occurs.
Depending on the type of data involved, the sampling distribution can be represented in a table format, as a histogram, or as a formula.
Sampling Distributions
There are essentially three things we want to know about the sampling distribution for any sample statistic:
What is the shape of the sampling distribution?
Where is the center (the mean) of the sampling distribution?
How much spread or dispersion (variation) does the sampling distribution have?
Sampling Distributions
Example: Imagine that we select 2 balls, with replacement, from a box containing two numbered balls and average the values that appear on the selected balls. One of the balls has the number 0 printed on it, and the other has the number 1 printed on it. In this scenario, what is the sampling distribution for the sample mean?
Let’s begin by listing all of the possible outcomes for the two selections. The possible outcomes are: 00, 01, 10, and 11. Next, we can determine each of the possible means:
Sampling Distributions
Because each ball has an equal probability of being chosen, each of the listed outcomes on the previous slide (00, 01, 10, and 11) has an equal chance of occurring (
Consider the table below:
Next, we will convert this table into a probability distribution for t he sample mean.
| Sample | P() | |
| 0,0 | 0 | 0.25 |
| 0,1 | 0.5 | 0.25 |
| 1,0 | 0.5 | 0.25 |
| 1,1 | 1 | 0.25 |
Sampling Distribution
| P() | |
| 0 | 0.25 |
| 0.5 | 0.5 |
| 1 | 0.25 |
Sampling distribution of the Sample Mean
Now that we have the probability distribution for the sample mean, we can use it to calculate the mean of the sample means and the standard deviation of the sample means:
Point Estimators
A point estimate is a statistic computed from a sample that is designed to estimate a population parameter.
The preferred estimate for the population mean () is the sample mean ().
So if we want to estimate the population mean, we would get some sample data and then we would determine the sample mean for the sample data, and that would be our point estimate.
The Standard Error of an Estimator
The Standard Error of an Estimator tells us how the estimator will vary from sample to sample.
The estimator will not be the same for every sample, so the standard error helps us understand how consistent the estimator will be from sample to sample.
Population mean:
Point Estimator:
Standard Error:
The Desired Traits of a Point Estimator
Ideally, our point estimators should be unbiased estimators.
Among unbiased estimators, we want the estimator with the minimum variance.
If an unbiased estimator is available they are preferred over biased estimators.
Example: Estimator A is not unbiased
because it misses almost always.
Estimators B and C are unbiased.
Estimator C has smaller variance than
Estimator B, hence it is called Minimum
Variance Unbiased Estimator (MVUE).
6.2 The Central Limit Theorem
The Central Limit Theorem
The Central Limit Theorem states that for a sufficiently large sample of size n, taken from a population that is not normally distributed, the sample mean has an approximately normal probability distribution.
In most cases, a sample size greater than thirty is large enough to assume that is approximately normal.
The Central Limit Theorem
The Central Limit Theorem describes the sampling distribution of the sample mean.
If all samples of size n are selected from a population of measurements with mean, , and standard deviation, , the distribution of the sample mean has the following mean and standard deviation (standard error):
Mean of the sample means is:
Standard error of the sample mean (the standard deviation of the distribution of sample means) is:
The Central Limit Theorem
If the population of measurements is normally distributed, the distribution of the sample means will be normal regardless of the size (n) of the sample.
However, if the population of measurements is not normally distributed, the distribution of the sample means will only be approximately normal when the sample size is suitably large.
As a good rule of thumb, we will assume that any sample size larger than 30 is large enough to ensure the distribution for the sample means is approximately normal.
This approximation will improve for larger values on n.
The Central Limit Theorem
Examples:
The random variable X has a highly skewed distribution. If samples of size 5 are taken from the population of X values, the distribution of the sample means will not necessarily be normal; however, if samples of size 35 are taken from the population, we can assume the distribution of the sample means will be approximately normal.
The random variable X has a normal distribution. If samples of size 2 are taken from the population of X values, the distribution of the sample means will be normal because the distribution of the sample means is normal at any sample size when X is normal.
The Mean of the Sample Mean
When discussing the Central Limit Theorem, which describes the sampling distribution of the sample mean, we stated that the mean of the sample means for all samples of size n is always equal to the population mean.
In other words, if all samples of size n are selected from a population of measurements with mean , the mean of the sample means is .
Example: If the true average IQ score for a population is 100 and every possible sample of size 15 is taken from the population, the sample means calculated from each of those samples will have an average equal to100 because that is the mean for the population.
For any particular sample size, the mean of all of the sample means is equal to the population mean.
The Mean of the Sample Mean
The Sample Mean IQ Scores for all Possible Groups of 15 People: (this is a partial list because the actual list would be very long)
To understand the idea discussed on the previous slide, imagine that for each sample of 15 individuals selected we calculate an average IQ score.
These sample means will be recorded (perhaps in a list like the ones illustrated above), and once we have calculated a sample mean from every possible sample of 15 people, we will then average all of those sample means in our list.
The result will be the population mean IQ score, which in this case is 100.
| 105 | 98 | 97 | 111 | 95 | 91 | 94 | 100 | 113 | 101 | 95 | … |
The Standard Error of the Mean
When we introduced the Central Limit Theorem, which describes the sampling distribution of the sample mean, we discussed the standard error of the mean (the standard deviation of the sample means) for all samples of size n.
In that discussion we stated that if all samples of size n are selected from a population of measurements with standard deviation , the standard error of the mean is .
Example: If the true standard deviation for IQ scores for a population is 15 and every possible samples of size 9 is taken from the population, the sample means calculated from each of those samples will have a standard error that is equal to
For any particular sample size, the standard error for the mean is equal to the population standard deviation divided by the square root of the sample size.
The Standard Error of the Mean
This definition of the standard error for the mean assumes that the sampling is done with replacement of that the population we are sampling from in infinite.
Sampling with replacement implies that a value that has been selected during the sampling procedure is available to be selected again and again in the same sample.
In the extreme case, this sampling procedure could produce a sample of n measurements which consists entirely of one value repeated n times.
This underlying assumption is a concern, because typically we do not take samples from infinite populations, and typically, we do not sample with replacement.
For example, if we are conducting a study on human height by measuring 10 randomly selected people, we probably would not want our sample to consist of one person’s height repeated 10 times.
The Standard Error of the Mean
Fortunately, we can modify the standard error formula to accommodate the finite population case pretty easily.
To find the standard error of the mean when sampling from a finite population, we use a multiplier often referred to as the finite population correction factor:
where N is the size of our population and we are selecting a sample of size n
If we are taking a sample of size n, without replacement, from a finite population of size N, the standard error for the mean becomes:
The Standard Error of the Mean
The formula for the standard error of the mean when sampling from a finite population only differs from our previous formula by the finite population correction factor., and often, we can ignore this difference.
When sampling from a large finite population without replacement, it is acceptable to use the original formula we provided as an approximation to the standard error of the mean.
How large does our finite population have to be to use this approximation?
Typically, if our sample size is not more than 5 percent of the population, we can use the population standard deviation divided by the square root of the sample size to approximate our standard error for the mean.
For the exercises included in this course, we have assumed that you will not be using the finite population correction factor. This means you can safely use the formula provided on the first slide when you are asked to determine the standard error for the mean.
The Variation in the Sample Means
If all samples of size n are selected from a population of measurements with standard deviation , the standard error of the mean is .
Because the standard error of the mean is equal to the standard deviation, σ, divided by the square root of the sample size, the standard error for the mean is always less than the standard deviation for the random variable.
The Variation in the Sample Means
This implies that a set of sample means from a population will also exhibit less variation than the random variable for that same population.
For example, if the sample size for the sample means is 4, the standard deviation for the sample means will be half as large as the standard deviation () for the random variable.
This means the distribution for the sample means is more clustered around the mean for the population than the distribution for the random variable is.
This is a useful trait because it implies that as the sample size increases, our sample means will move closer and closer to the true population mean.
The Variation in the Sample Means
Example: An investor has two sets of data involving the closing stock price for a company in the NASDAQ. One set of data contains the closing stock prices for a random selection of 12 days taken over the course of the year, and the other set of data contains 12 averages obtained from random samples of 4 days of closing prices taken over the same year. Which data exhibits a larger amount of variation?
The Variation in the Sample Means
Closing prices of 12 randomly chosen days (sample standard deviation s = $85.24):
Average closing prices for 12 samples of n = 4 days (sample standard deviation s = $38.89):
It is clear that the set of averages has far less dispersion than the set of individual observations.
If we had every sample mean possible for all samples of four (selected with replacement), the standard deviation of these sample means would be , where is the standard deviation for the daily closing prices.
The Central Limit Theorem and Calculating Probabilities for the Sample Mean
A software company’s average daily stock price last year was $38.12. The standard deviation for those prices was $2.45. If a random selection of 32 days were chosen from last year, what is the probability that the average price of the company’s stock for those 32 days is more that $37.00?
By Central Limit Theorem, since the sample size (32) is greater than 30, the distribution of the average stock prices is approximately normal. So
The z-score for $37 is:
.
So
