Lecture Notes

Chapter 7 Samples, Sampling Distributions, and Confidence Intervals

We begin the transition from descriptive to inferential statistics here. Samples, sampling, and the techniques for making inferences about points and intervals. We will trade off some certainty when we make such inferences, but we will, at least, be able to say how confident we are about our inferences. We will also be able to gain much time, most of that saved in the data collection process, because we will be dealing with samples not populations. THIS CHAPTER IS THE KEY TO THE REST OF THE TEXT. (Revised 2003-07-30)

Samples and Sampling
- Do you remember what a sample is? This is a good time to review.
- A sample is any group selected from a population for study by a researcher. However, the power of samples differs markedly depending on the type and size of the sample chosen. Some classic mistakes in prediction are examples of faulty sampling. Non-random samples will usually not lead to accurate prediction. However, random samples help prediction tremendously. A random sample occurs when every member of a population has an equal chance of being made a member of the sample. Large random samples help control against selection, a form of bias. Other types of samples also help make better predictions, they include matched samples and stratified random samples.
- Famous Sampling Errors (Biased Samples)
  - Presidential Elections
  - 1970 Draft Lottery
- Types of Samples
  - Nonprobability samples
  - Probability samples
    - simple random samples--draft lotteries
    - biased samples--some members of the population are systematically excluded
  - Discussion of Samples
  - Research Samples
    - Usually not random
    - Random Assignment
      - Widely used process for creating research groups
- Costs and Benefits of Sampling
  - Variable results will occur (uncertainty)
  - Sampling distribution, however, will provide a measure of the uncertainty
  - If uncertainty is low, then sampling is a good way to get answers
  - If uncertainly is high, then you can now say, "I am uncertain about these results."
- Randomness
  - random means that no pattern can be observed in a set of observations
  - randomness is usually achieved via some random procedure like shuffling cards, using a random number table, or using a computer
  - to use the random number table (Table B) enter it at random, then using some previously agreed upon pattern of movement, pick numbers (Can you see why the pattern of movement must be decided upon before?)
  - pick numbers to fill up a previously made data sheet, disregard any numbers which are out of range, or which you have already used
  - Generating Random Samples
Sampling Distribution of the Mean
- From a large population, take many samples. Then, from each such sample, compute the sample mean. The distribution of those sample means is the sampling distribution of the mean.
- The mean of the means (got that?) is a good estimate of the population mean from which those means were drawn.
- The mean of the means is called the expected value
- The standard deviation of the means is called the standard error
- Get this, the shape of the sampling distribution of the mean will be normal, but that is not all, see below

Central Limit Theorem

If a population distribution has a mean m and a standard deviation s, then the distribution of sample means drawn from this population approaches a normal distribution with a mean of m and a standard deviation of:
Furthermore, the mean of the sampling distribution approaches m, the population mean, as N increases.
For non-symmetrical distributions, the sampling distribution will be normal when N > 30
For symmetrical distributions, the sampling distribution will be normal when N < 30
All of this true even if the original distribution IS NOT NORMAL!
Relationship of standard error and the standard deviation (Problem 11, page 146)

Relationship of Standard Error, Standard Deviation, and N

N s = 1 s = 2 s = 4 s = 8

1
1.000

2.000

4.000

8.000

4
0.500

1.000

2.000

4.000

16
0.250

0.500

1.000

2.000

64
0.125

0.250

0.500

1.000

As N increases the standard error of the mean decreases

t distribution
- A FAMILY of curves for each sample size or N
- Degrees of Freedom
  - the t distribution uses 0 as its mean and the area always is 1, so for t, the degrees of freedom (df) will be N - 1, where N is the number of observations
  - other statistics will also use the concept of degrees of freedom, but those dfs may be calculated differently
  - dfs are necessary to keep the theoretical distribution values constant (the last variable must be fixed)
- W. S. Gosset discovered t distribution and used it first (Publishing as "Student")
- for large N (>30), t approaches the normal distribution (see p. 149)
- use t when you cannot meet the assumptions required to use standard error as described above (large samples and random selection)
- compared to the normal curve and for small samples, t :
  - has fewer observations in the center of the distribution
  - has more observations in the tails of the distribution
- use t when sample sizes are small and/or s is unknown
- Table D (p. 368) shows values for t distribution
- When the df for t distribution is equal to infinity, t is the same as z
Confidence Intervals
- One of the uses of inferential statistics is point estimation. In other words, estimating how confident we can be that a sample mean truly represents the population mean.
- While we can never be certain, we can specify just how certain we are about our point estimation
- The means to that end is the confidence interval
- Formula for confidence interval (using z) for t substitute appropriate t value for z from Table D (see example below):
- Logic of confidence interval
  - We know that any sample mean is an estimate of the population mean
  - By using + or - 2 SD units from the sample mean, we will include 95.44% of all of the sample means possible
  - By using + or - 3 SD units from the sample mean, we will include 99.74% of all of the sample means possible
  - Instead of 2 or 3 SD units, we use the z score values for 95% and 99%, which are: 1.96 and 2.58, respectively (it is just neater and cleaner)
- Example problem (number 21, page 151):
  For the following ficticious data about a sample of lightbulbs: N = 16, SX = 15,600, SX² = 15,247,500, calculate the mean, standard deviation (s hat), and the standard error of the mean.
Calculate the mean by dividing SX = 15,600 by N = 16. That gives a mean of 975 hours.

Use the formula below to calculate the standard deviation (s hat)

s hat = square root of:15,247,500 - (15,600)² / 16 / 15

s hat = square root of: 15,247,500 - 15,210,000 / 15

s hat = square root of: 37,500 / 15

s hat = square root of: 2500

s hat = 50

Now use formula below to get the standard error of the mean:

standard error of mean = 50/ square root of 16
standard error of mean = 50/ 4
standard error of mean = 12.5
- Then for the 95% confidence interval, using the formulas above, find t value for 95% confidence interval from Table D for 15 df (N-1). That value is = 2.131
  LL = 975 - 2.131 (12.5) = 948.36 hours
  
  UL = 975 + 2.131 (12.5) = 1001.64
  
  For the 99% confidence interval, find 99% confidence interval from Table D for 15df (N-1). That value is 2.947
  
  LL = 975 - 2.947 (12.5) = 938.16 hours
  
  UL = 975 + 2.947 (12.5) = 1011.84 hours
  
  1000 hours is inside both confidence intervals (barely)
- Example problem (number 22, page 154)
Fourteen social workers took an eight week course in assertiveness. The national mean for assertiveness on the Door Manifest Assertiveness Test is 24.0. Using the following data, construct the 95% confidence interval for the sample mean. Data:

SX = 378, SX² = 10,400

Solution:

Mean = 378/14 = 27

= 3.863

s_x = s(hat)/ ÷N = 1.032

Then, t = 2.160 for 13 df, so:
LL = mean - t (s_x) = 24.77

UL = mean + t (s_x) = 29.23
Note that the 95% confidence limit exceeds the population mean, thus the workshop was effective at raising assertiveness

URLs
- Sampling
  - Random number generator--demo of six random numbers/time (Brock U., CA) not working 2003-07-28
- The Central Limit Theorem
  - The Central Limit Theorem--NEW, interactive, basic, short, graphics
    - Allows users to "roll" up to 5 dice up to 10,000 times in order to see the shape of the sampling distribution. http://www.stat.sc.edu/~west/javahtml/CLT.html (works 2003-07-28)
  - Central Limit Theorem (1 of 2)--(Rice) (works 2003-07-28)
  - Quincunx--animated demo of central limit theorem (works 2003-07-28)
  - Central Limit Theorem Dice Demo--rolls numbers of dice specified times (SC) (works 2003-07-28)
- Sampling Distribution of the Mean
  - Sampling Distribution (1 of 3)--(Rice) (works 2003-07-28)
  - Area Under the Sampling Distribution of the Mean (1 of 4)--(Rice) (works 2003-07-28)
  - Standard Error of the Mean (1 of 2)--(Rice) (works 2003-07-28)
- t distribution
  - Critical Values of the t distribution--(UBC, CA) (works 2003-07-28)
- Degree of Freedom
  - Degree of Freedom--definition (Principia Cybernetica Web) (works 2003-07-28)
Quiz

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N	s = 1	s = 2	s = 4	s = 8
1	1.000	2.000	4.000	8.000
4	0.500	1.000	2.000	4.000
16	0.250	0.500	1.000	2.000
64	0.125	0.250	0.500	1.000