Chapter 7 Sampling and Sampling Distributions

Recall from before that the population is the set of all elements in a study while a sample is a subset of the population.

We also talked about statistical inference, which is when we develop estimates of the population from sample data and infer what the population must look like. This is done b/c:-Population data is generally not something you can obtain-if the sampling is good it is much quicker and easier to get the estimate and it should be reliable

A. Terms & Types of Simple Random Samples

1. Parameter- the number and variable that describes the population. In general we assume this to be unknown, but in some instances it is known.

2. Statistic – a value that is computed from the sample data. It comes exclusively from sample data and is not composed of any unknown parameters.

3. Point Estimation – this is using the data from the sample to compute the value of a sample statistic. This is what serves as an estimate for the population parameter.

a. point estimator – the estimate of the population parameter; an example is _

x a point estimator for μ.b. point estimate – the actual value that is computed for the point estimatorex:

_

x = 4.7c. sampling error – the absolute value of the difference between the point estimate and the actual population statistic.Formula: | point estimate – population parameter |

4. Simple Random Sample (Finite Population) – a SRS of size n from a finite population so size in is selected from all possible samples of size n. In this case each has an equally likely probability of being selected.

-Two ways to do this:(a) Sampling with replacement – choose the sample and once a piece of data is chosen for your sample you take it out of the population(b) Sampling w/o replacement – in this instance after a piece of data is chosen for a sample, it is put back into the population and may be chosen again at random.

5. Simple Random Sample (Infinite Population) – in this instance there are an infinite number of population data points. To be considered an infinite population SRS it must satisfy the following two conditions:

1

(a) Each element must come from the population specified no misidentification of population.(b) Each element is selected independently

B. Introduction To Sampling Distributions

-if you continue to take samples of data and compute every possible combination of samples (i.e. all permutations or combinations) of size n then the sample statistics/point estimators can have their own distribution. -so each sample statistic/point estimator will have its own distributions with its own mean, variance, and standard deviation.-we we know what type of distribution this is we can make probability statements from it and assess how close the point estimates are to the population parameters (i.e. how close _

x is to μ)

1. Sampling Distribution – the probability distribution of any particular sampling statistic.

2. Law of Large Numbers – if we draw observations from a population with a finite mean μ at random, as we increase the number of observations we draw the value of the sample mean (

_

x ) gets closer and closer to the population mean.

-note that this makes sense b/c as you increase the size of your sample it gets closer to the size of the population. So it begins to look more and more like the population itself. For this reason the mean should approach the population mean.

3. Sampling Distribution of _

x - this is the probability distribution of all possible values of a sample mean given a certain size sample n.

Ex:

-Suppose we have a distribution as follows:If we want to create a sampling distribution wewould take samples of size n, let’s say 15from the distribution to the right and fromeach sample obtain a mean, variance,and standard deviation.

2

x

15 20 25

Sample1-has own_

x , s2, &

s

Sample 2-has own_

x , s2, &

s

…..continue with process for all possible samples of size 15. If we do this we can take the values from each sample and create its own distribution as shown below.

Graphically:

-notice now we have a distribution of sample means.This distribution is created from the means of each sample and it has its own variance and standarddeviation. Note that they should be muchsmaller than the distribution sampledfrom since we created it from the samplemeans of the data.

4. Characteristics of the Sampling Distribution of _

x

a. E (_

x ) = μ so the mean of all values of _

x should be the population mean μ. This is called unbiasedness. Since the value of the sampling statistic converges to the population parameter.b. Standard Deviation of

_

x - called the standard error of the mean it tells us how close our estimates of the mean are to the actual mean.i. finite population value – σ _

x = )1/()( −− NnN * (σ / n )

ii. infinite population - σ _

x = σ / n

note: σ = population variance, N = population size, n = sample size; must still use the infinite population estimate if n/N < 5% of the population size.

5. Central Limit Theorem – when choosing n and it is a SRS we can assume that the sampling distribution of

_

x ~N as N gets larger and larger. If it is greater than 30 we assume it is Normal. -if the population is normal, then the sampling distribution must be normal and this rule does not apply. This is for any size of sample.-as n increases the variance and standard deviation get tighter and there is a higher probability that the sample means is within a certain distance of the actual population mean.

3

_

x

20

Image 1: Seeing how the sampling distribution changes shapes from 1, 2, 10 and finally 25 observations.

6. Statistical Process Control and _

x Control Charts

-goal of statistical process control is to make a process stable or controlled over time. It does not mean that there is no variation; just that it is much smaller in magnitude over time.

a. In control – when a variable can be described by the same distribution when observed over time.

b. control charts – tools that monitor a process and alert us when a process has been disturbed. It is said to be ‘out of control’ when it does this.

c. _

x control charts or_

x -chart - these can be used to monitor whether or not a process is staying within some upper and lower bound that the tester designates. To do this you would draw a horizontal line at the mean and then find the upper and lower bound with the following formulas:upper bound: μ + z * σ / n

lower bound: μ - z * σ / n

4

Note: that the tester determines how far away is acceptable in this process. It could be 3 standard deviations (z’s) or it could be less. It depends on what is designated as a stable process over time.

Graphically:

Note: as long as the sample points stay within the red-lines the process is ‘in control.’ As soon as you obtain a measurement that puts it outside of the upper and lower bounds the process has been disturbed somehow and needs to be adjusted to put it back on a steady path.

7. Unbiasedness and Minimum Variance Estimates

a. Unbiasedness – In general we say that a sampling statistic in unbiased if its sampling distribution value converges to the population value.i.

_

x μ so it is an unbiased estimator.

ii. s2 σ2 so it is also an unbiased estimator.b. Minimum variance estimate – since the sampling distribution of

_

x produces the smallest variances estimate of all possible other values that could estimate the mean (like median, mode, or any estimator). So we way it is MVE or the minimum variance estimator.

C. Inference about a Population Proportion

Now we are concerned with finding out and exploring proportions. Many of the techniques and statistics that we have used in previous chapters will be used again. So it should seem very familiar how we go about studying and analyzing this type of procedure. Just make sure to note the definition of a proportion below.

1. Proportion – this is the percentage that our population takes on a certain characteristic.

p = number of successes / total individuals

5

μ

sample

Lower bound

Upper bound

Distance = -z * σ / n

= this is the sample proportion and is designated at p-hat. It is an actual calculated

value.

Example: Number of students who passed a class out of 20. If we let passing be greater

than 70% and we find that 14 students had scores greater than 70% then our is:

= 14 / 20 = 0.70

2. Sampling Distribution of p:

Just as with the mean we had a sampling distribution that had certain characteristics, we can also note that p also has the following characteristics.

a. The expected value or mean of the sampling distribution is p (i.e. the population proportion)

so E( ) = p

b. The standard deviation of p or =

c. graphically: We can use this just as before with our z. So if we assume that we have a normal distribution with a large enough sample size then our Z becomes:

Z = ) /

So if we were given that p = 0.60 and n = 36 and wanted to know the P ( . We

can calculate our Z = (0.53 – 0.60) / 0.0816 = -0.86 where = ≈0.0816.

So this is the same as asking P (Z < -0.86) = 0.1949

6

Z

So from our Z-table we find this value is 0.1949 or about 19.49%

-0.857

Z

= this is the sample proportion and is designated at p-hat. It is an actual calculated

value.

Example: Number of students who passed a class out of 20. If we let passing be greater

than 70% and we find that 14 students had scores greater than 70% then our is:

= 14 / 20 = 0.70

2. Sampling Distribution of p:

Just as with the mean we had a sampling distribution that had certain characteristics, we can also note that p also has the following characteristics.

a. The expected value or mean of the sampling distribution is p (i.e. the population proportion)

so E( ) = p

b. The standard deviation of p or =

c. graphically: We can use this just as before with our z. So if we assume that we have a normal distribution with a large enough sample size then our Z becomes:

Z = ) /

So if we were given that p = 0.60 and n = 36 and wanted to know the P ( . We

can calculate our Z = (0.53 – 0.60) / 0.0816 = -0.86 where = ≈0.0816.

So this is the same as asking P (Z < -0.86) = 0.1949

6

Z

So from our Z-table we find this value is 0.1949 or about 19.49%

-0.857

Z