1. Chapter 5 Review
Probability – the relative likelihood of occurrence of any given outcome or event, ranges from 0 to 1
Converse Rule (not)
Multiplication Rule (and)
Addition Rule (or)
Z Scores – Indicates direction and degree that any raw score deviates from the mean in sigma units
Values for one side of the normal curve given because of symmetry
Use Table A (p. 377)
Column B for area between the raw score and the mean
Column C for area beyond the raw score (to the tails)
Beginning of “inferential statistics” in order to generalize from samples to populations
µ = mean of a distribution
σ = standard deviation of a distribution
X = raw score
Z = standard score

2. Probability
There are 10 people randomly selected: 7 males and 3 females. Assume that 60% of the population has brown hair, 25% has black hair, and 15% has a different color; “other.”
What is the probability that we will select...
a male? – a male with black hair?
not a female? – a female with brown hair?
two males? – a female w/brown hair or a female
one of each gender? w/black hair?
two females? – a male w/brown hair and a male w/black hair?
a male w/brown hair and not a female w/brown hair?

3. y

4. Probability: Mean 28; Std Dev 4
Z = (33-28) / 4 = 5 / 4= 1.25

5. Chapter 6 Samples and Populations

6. Introduction
Often, there is no capacity to measure an entire population
We are limited by time, energy, and economic resources
Sampling allows researchers to generalize to the wider population and has become an integral part of social science research

7. Samples and Populations
Set of individuals who share at least one characteristic
Sample:
A smaller number of individuals from the population that share one or more given characteristics

8. Sampling Methods Researchers want to make inferences from data.
Sampling methods vary according to population and access.
Two types of sampling methods:
Non-random samples
Random samples

9. Nonrandom Sampling
This is where each member of the population has an unequal probability of selection
Done when:
You cannot obtain a probability sample
A topic or study setting is inappropriate for probability sampling

10. Nonrandom Samples 4 main types of nonrandom sampling:
Convenience – select units that are available
Quota – sample drawn in proportion to the population
Purposive/judgment – logic, common sense, or judgment used to select a sample that is presumed representative of a larger population
Snowball – find one person, ask if they known anyone that would like to participate, etc.

11. Random Sampling
This is where every member of the population has an equal chance of being drawn into the sample
Therefore, every member must be identified
Simple random sampling – a table of random numbers is employed to select a sample that is representative of a larger population
Variations
Systematic – every nth member of a population is included in the sample
Stratified – the population is divided into homogeneous subgroups from which simple random samples are drawn
Cluster/multistage – sample member are selected on a random basis from a number of well-delineated areas known as clusters (or primary sampling units)

12. Sampling Error Mean of a sample shown as
Mean of a population shown as µ
Standard deviation of a sample shown as s
Standard deviation of a population shown as σ
Mean or standard deviation of a sample rarely identical to the population
This difference is known as sampling error.

13. Table 1: A population and three random samples of final exam grades70 80 93 86 85 90 56 52 67 40 78 57 89 49 48 99 72 30 96 94
μ = 71.55
Sample A
Sample B
Sample C 96 40 72 99 86 56 49 52 67
= 75.75
= 62.25
= 68.25
N = 20

14. Sampling Distributions of Means96 93 92 98 96
100
106
102
105
103 99
107
101 91
102
108
104 95
Figure 1: The Mean Long Distance Phone Time in 100 Random Samples in
Which the Mean = min

15. Characteristics of a Sampling Distribution of Means
The sampling distribution of means approximates a normal curve.
The mean of a sampling distribution of means (the mean of means) is equal to the true population mean.
The standard deviation of a sampling distribution of means is smaller than the standard deviation of the population.
-The sample mean is more stable than the scores that comprise it.

16. The Sampling Distributions of Means as a Normal Curve
Most sample means will fall close to the true population mean—within 3 standard deviation units above and below µ
Standard deviation of sampling distribution is denoted
Shows theoretical probability, standard deviation among all possible sample means
Allows a new formula for z with samples:
o signifies that it is a theoretical probability distribution and the X signifies that this is the standard deviation among all possible sample means

17. The Sampling Distribution of Means as a Normal Curve
In Chapter 5, we discussed probability in terms of the likelihood of occurrence
Normal curve = probability distribution
For instance, to find the probability associated with obtaining someone with an hourly wage between $10 and $12 in a population having a mean of $10 and a standard deviation of $1.5,
We translate the raw score $12 into a z score (+1.33) and go to Table A in Appendix B to get the percent of the distribution.

18. Standard Error of the Mean
Standard deviation of theoretical sampling distribution can be derived.
This is known as the standard error of the mean.
Formula for standard error of the mean:
We can now calculate the range of mean values in which our population mean is likely to fall.

19. Standard Error of Mean
Obtained by dividing the population standard deviation by the square root of the sample size
Illustration: IQ test
Population mean of 100
Population standard deviation of 15
If we took a sample of 10, subject to a standard error of?
With the aid of the standard error of the mean, we can find the range of mean values within which our true population mean is likely to fall.

20. Confidence Interval Cont.
Can be constructed for any level of probability
It is has become a matter of convention to use a wider, less precise confidence interval having a better probability of making an accurate or true estimate of the population mean.
68% confidence interval = ± (1)
95% confidence interval = ± (1.96)
99% confidence interval = ± (2.58)
1.96 is used as it covers 95% of the distribution (+/- 2 std dev)
2.58 is used as it covers 99% of the distribution (+/- 3 std dev)

21. Total Area under the Normal Curve

22. Confidence Interval Cont.
How do we go about finding the 95% confidence interval?
We already know that roughly 95% of the sample means in a sampling distribution lie between -2 standard deviations and +2 standard deviations from the mean of means.
Going to Table A, we can make the statement that 1.96 standard errors in both directions cover exactly 95% of the sample means (47.50% on either side of the mean of means).
Plus/Minus 1.96
If we apply the 95% confidence interval to our estimate of the mean IQ of a student body, we see that 95% confidence interval = (1.96)(3) =
= to

23. Confidence Interval Cont.
An even more stringent confidence interval is the 99% confidence interval.
From Table A, we see that the z scores 2.58 represents 49.50% of the area on either side of the curve.
Doubling this amount yields 99% of the area under the curve;
99% of the sample means fall into this interval
By formula:
99% confidence interval
= Sample Mean standard error of the sample mean

24. 95% Confidence Interval Using z
Suppose we want to determine the expected miles per gallon for a new Ford Explorer?
Standard deviation = 4 miles/gallon
N = 100 cars
Sample Mean = 26 miles/gallon
How do we obtain a 95% confidence interval for the mean miles/gallon for all cars of this model?
Step 1: Obtain the mean for a random sample
Step 2: Calculate the standard error of the mean
Step 3: Compute the margin of error
Step 4: Add and subtract the margin of error from the sample mean

25. 99% Confidence Interval Using z
Now, the statistician is informed that 95% confidence is not confident enough for their needs. To be confident, we want 99%.
Standard deviation = 4 miles/gallon
N = 100 cars
Sample Mean = 26 miles/gallon
Step 1: Obtain the mean for a random sample
Step 2: Calculate the standard error of the mean
Step 3: Compute the margin of error
Step 4: Add and subtract the margin of error from the sample mean

26. End Day 1

27. The t ratio
Very few situations in which the population standard deviation (and thus the standard error of the mean) is known
When the exact standard deviation of the population (σ) is unknown, the t-distribution is used
Recall that sample means (and their standard deviations) are lower and more stable than population means
It is then necessary to inflate the sample standard deviation to produce more accurate estimates
Standard Error for a t ratio

28. The t ratio Cont.
Still, there are two very different purposes for calculating the variance and standard deviation:
(1) to describe the extent of variability within a sample of cases or respondents and
(2) to make an inference or generalize about the extent of variability within the larger population of cases from which a sample was drawn.

29. Degrees of Freedom
The greater the degrees of freedom, the larger the sample size and the closer the t distribution gets to the normal distribution
df = N – 1
Recall that the only difference between a t and a z is that the former uses an estimate of the standard error based on sample data while the latter is known

30. Cont.
What would one do for larger samples for which the degrees of freedom may not appear in Table B?
A sample size of 50 produces 49 degrees of freedom
t value for 49 df and alpha = .05
40 df = 2.021
60 df = 2.000
Use the fewer degrees of freedom (in this case, 40)
CI = Sample mean +/- the T value multiplied by the standard error
CI = ± t*sx
T values get smaller and tend to converge as degrees of freedom increase. Starts at and approaches a limit of Go with a more conservative test to ensure you do not commit a Type 1 error (accepting the null hypothesis when it’s in fact false).

31. The t ratio For t-distributions, use Table C instead of Table A
Various levels of alpha
Alpha = area in the tails of the t distribution
For a 95% level of confidence, an alpha = .05.
For a 99% level of confidence, an alpha = .01.
With the addition of alpha, we now have two pieces of information available and can now construct our confidence interval
Degrees of freedom (N – 1)
Alpha value (95% = .05 or 99% = .01) Example: Construct a 95% CI with a sample of 20 respondents, a mean of 40, and a standard deviation of 3.5.
df = 19, alpha = .05
t = 2.093

32. Confidence Intervals
Probability that our population mean actually falls within the range of mean values (within which our true population mean is likely to fall)
Can be constructed for any level of probability
It is has become a matter of convention to use a wider, less precise confidence interval having a better probability of making an accurate or true estimate of the population mean.
Most researchers and media surveys use a 95% CI. A more conservative test is the 99% as it sacrifices precision in order to be more confident.

33. An Illustration
Suppose that a researcher wanted to examine the extent of cooperation among kindergarten children. To do so, she unobtrusively observes 9 children at play for 30 minutes and notes the number of cooperative acts engaged in by each child: The mean number of cooperative acts was 2.67 and the standard deviation was 1.32.
Step 1: Obtain the estimated standard error of the mean
Step 2: Determine the value of t from Table C
Step 3: Obtain the margin of error by multiplying the standard error of the mean by the t-ratio
Step 4: Add/Subtract this product from the sample mean to find the interval within which we are 95% confident the population mean falls

34. Estimating Proportions
We can also estimate population proportions.
Pattern of formulas is the same.
sP = standard error of the proportion
P = sample proportion
N = total number in the sample
We either use:
95% CI = P +/- 1.96*sp or 99% CI = P +/- 2.58*sp

35. An Illustration
Suppose a polling organization contacted 400 members of a local police union and asked them whether they intended to vote for candidate A or candidate B. Suppose that 60% reported their intention to vote for candidate A. Find the 95% confidence interval for candidate A.
Step 1: Obtain the standard error of the proportion.
Step 2: Multiply the standard error of the proportion by 1.96 to obtain the margin of error.
Step 3: Add and subtract the margin of error to find the confidence interval.

36. Summary Researchers rarely work with entire populations.
Sampling is necessary.
Sampling errors occur normally.
Mean of sampling means equals true population mean.
We can estimate standard deviation of sampling distribution of means.
Confidence intervals for means (or proportions)
t distributions introduced