Basic Quantitative Methods in the Social Sciences (AKA Intro Stats)

1
Basic Quantitative Methods in the Social
Sciences(AKA Intro Stats)

• 02-250-01
• Lecture 5

2
Sampling Distributions

• Inferential Statistics generalizes findings
  obtained from samples to the populations that the
  samples were drawn from
• Samples need to be representative of the
  populations they are drawn from so we use
  random sampling

3
Random Sample

• Random Sample a sample in which each member of
  the population has an equal chance of being
  included
• We cannot assume that a random sample is exactly
  representative of its population
• E.g., randomly choosing 50 students from this
  class their mean age may not be exactly the
  mean age of the entire class (the population
  approx 230 students)

4
Random Sampling

• Random sampling makes all the samples which could
  be drawn from the population equally likely
  (e.g., who is included in the 50 student sample)
• Each of the possible samples of 50 students would
  have mean ages that would slightly differ from
  the population mean age
• We measure this difference with sampling error

5
Sampling Error

• Sampling Error the difference between a
  statistic and the parameter it estimates
• E.g., if the population mean age was 24 and the
  sample mean age was 21, we say we have a sampling
  error of 3 years

6
Sampling Error

• Because we usually dont collect data for an
  entire population, we must have some way of
  estimating the sampling error size and account
  for it when we generalize sample information to
  populations
• We often obtain more samples to determine the
  sampling error

7
Sampling Distributions

• If we draw 6 samples of 50 students from this
  class, we can obtain a better estimate of the
  true population mean age than if we only drew one
  sample
• Suppose the mean ages for those 6 samples were as
  follows
• 25, 23, 23, 25, 25, 26
• The mean of these 6 mean ages is 24.5

8
Sampling Distributions

• Looking at the mean age of the first sample, 25
  years, if we only had data for this one sample,
  25 years would be our best estimate of the true
  population mean
• By taking more than one sample, we calculate a
  more accurate estimate of the population mean,
  24.5 years

9
Sampling Error

• Since all of the ages are relatively close to
  each other, we can say with greater certainty
  that we have small sampling error for any one of
  the sample means
• If the samples mean ages were much more
  dissimilar, any one of the sample age means would
  probably have a much higher sampling error

10
Sampling Error

• This means that the variability of a statistic
  over repeated samplings gives us some indication
  of sampling error
• If we continued to draw samples from the
  population until all possible samples had been
  drawn and the statistic of interest (mean age) is
  entered into a frequency distribution, this is
  known as a sampling distribution

11
Sampling Distributions

• Sampling Distribution the distribution of a
  statistic over repeated sampling from a specified
  population
• Using our previous example, the sampling
  distribution of the mean for this class is a
  distribution of the means of every possible
  sample of 50 students

12
Expected Value

• The mean of a sampling distribution of
• is known as the expected value of the mean
  the mean of sampling means
• We use the symbol ? instead of for the mean
  of a sampling distribution because it is a
  population of terms

13
Standard Error

• The standard deviation of a sampling distribution
  is know as the standard error (?x) the
  standard amount of difference between and ?
  that is reasonable to expect simply by chance
• The mean of any sample we take can be plotted on
  the sampling distribution of X if we know the ?x
  and ?x
• The sampling distribution of X is a normal
  distribution

14
Sampling Distribution
Sampling error
?x ? x
Obtained from one sample
15
Standard Error

• The formula for standard error is as follows

16
Sampling Distributions

• We usually dont know ?x and ?x and must
  estimate ?x
• Sampling Distributions are the basis for many
  statistical tests (e.g., t-test well talk
  about this later)
• Statistical tests are a mathematical way of
  testing a hypothesis

17
Hypothesis Testing

• Hypothesis testing is a way of examining a
  statement about a relationship between
  independent and dependent variables
• Independent variable the variable whose effects
  the experimenter is interested in studying
• Dependent variable the variable that the
  experimenter measures (the data)

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Independent and Dependent Variables - Example

• If an experimenter is interested in researching
  how hours of studying for an exam affect
  performance on a test, the variables are as
  follows
• Independent Variable (IV) hours spent studying
• Dependent Variable (DV) performance on test
  (e.g., grade received)

19
Independent Variables

• There are 2 broad types of IVs
• Treatment Variable a treatment the experimenter
  applies to previously undifferentiated
  participants
• E.g., certain participants are told to study for
  5 hours and others are told to study for 2 hours
• Categorical Variable A characteristic that is
  inherent to, or pre-exists, in the participant
• E.g., gender you cant assign someone a gender

20
Levels of IV

• We also talk about the levels of IVs how we
  break down the IV
• E.g., if we are interested in studying the IV of
  hours spent studying, it could have 2 levels 2
  hours and 5 hours
• Studying the IV of gender has 2 levels male and
  female
• The levels of an IV are compared on their DV
  scores to look for a difference in outcome is
  there a difference in test performance between
  those who study for 5 hours and those who study
  for 2?

21
Time to Think

• A nursing researcher wants to know if giving TLC
  prolongs life in cancer patients. 50 cancer
  patients are divided into two groups group A
  (n25) is given TLC by their nurses, and group B
  (n25) are not. What is the DV, IV, and levels of
  IV?
• A researcher wants to know if members of the
  Federal Liberal Party are wealthier than are
  members of the Federal NDP. 100 members of each
  party are asked to submit financial statements.
  What is the DV, IV, and levels of IV?

22
Null Hypothesis

• Tests of hypotheses in science are decisions to
  retain or reject a null hypothesis (Ho)
• Null hypothesis (Ho) a statement of
  relationship between the IV and DV, usually a
  statement of no difference or no relationship
  we assume there is no relationship between IV and
  DV

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Null Hypothesis Examples

• Men and women do not differ in IQ (?men ?women)
• Hours spent studying do not affect test
  performance (?2 hours ?5 hours)
• Height does not affect weight (?short
  ?tall)

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Null Hypotheses

• Null hypotheses contain 3 components
• The IV comparison being made
• The DV being measured
• The null relationship between IV and DV (e.g.,
  do not differ)

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Alternative Hypothesis

• Although not directly tested, the Alternative
  Hypothesis (Ha) does state a relationship, or
  effect, of the IV on the DV this is often
  called the Research Hypothesis
• E.g.,
• Ha Men and women do differ in IQ (?men ?
  ?women)
• Ha Women have higher IQs than men (?women gt ?men)

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Directional Ha

• Ha Women have higher IQs than men (?women gt
  ?men) is a directional alternative hypothesis
  we state that one level of the IV will have
  greater (or lesser) DV scores than the other
  level
• When we make a directional alternative
  hypothesis, we have a reason (either based on
  past research or a theory) to predict the
  direction of the results (i.e., that a statistic
  at one level of the IV will be greater or less
  than the statistic at the other level of the IV)
  (note the above example is hypothetical only)

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Non-Directional Ha

• A non-directional alternative hypothesis does not
  state the expected direction of effect
• Ha Men and women have differing IQs (?women ?
  ?men)
• We make a non-directional alternative hypothesis
  when we have no reason to predict the direction
  of the results. For instance, since there is no
  theory or research body that would suggest that
  women should have higher IQs than men, we would
  only predict that their IQs are different than
  mens

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Hypothesis Testing

• Hypothesis testing looks at the observed
  difference in DV scores between the levels of the
  IV and compares this difference to the expected
  difference (Ho)
• Any difference in value of the DV between the
  levels of the IV can be explained in 2 ways the
  effect of the IV or sampling error

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Hypothesis Testing

• Testing the null hypothesis is a way of
  determining the probability that the observed
  outcome could be found if the null hypothesis was
  true
• E.g., if we did find a difference between the IQs
  of men and women, what is the chance we would
  find this result if there is actually no
  difference between their IQs?

30
Confidence Levels

• When this probability drops below a certain
  level (a criterion level), we call the result
  significant
• This criterion level is known as the confidence
  level of the test, or alpha (?)

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Confidence Level

• Confidence Level a criterion level of
  probability (alpha ?), set by the experimenter,
  which acts as the reference for deciding whether
  to reject or retain the null hypothesis
• Significant Result at .05 we determine the null
  hypothesis is not true but there is a 5 chance
  that the null hypothesis is actually true.

32
Confidence Level

• The confidence level is set by the experimenter,
  but generally the convention is to use ? 0.05
  and ? 0.01
• For ? 0.05, this means that there is a 5
  chance we will reject the null hypothesis when it
  is actually true

33
Rejecting the Null Hypothesis

• If the likelihood of observing this outcome is
  below the confidence level (? 0.05 or ?
  0.01), then we say that the result is significant
  and we reject the null hypothesis
• Significant results reject Ho (there is a
  difference)
• Non-significant results retain Ho (there is no
  difference)

34
Type I and Type II Errors

• When we decide to retain or reject the null
  hypothesis, we never do so with 100 certainty we
  are making the right decision we make the
  decision with a probability of being correct (the
  alpha level)
• We can make an incorrect decision, resulting in 2
  types of errors, Type I or Type II

35
Type I Errors

• Type I Error Rejection of the null hypothesis
  when it is true
• We conclude that the IV affects or is related to
  the DV when in reality the result was due to
  sampling error
• We see something that is not really there

36
Type I Error Example

• If our null hypothesis is that men and women do
  not differ in IQ, the Type I error is
• Finding a result that men and women do differ in
  IQ, when in reality they do not
• We find this difference because of sampling error

37
Type II Errors

• Type II Error Retention of the null hypothesis
  when it is false
• We conclude that the IV does not affect or is not
  related to the DV when in reality there is an
  effect or relationship
• We fail to see something that is really there

38
Type II error Example

• If our null hypothesis is that men and women do
  not differ in IQ, the Type II error is
• Finding a result that men and women do not
  differ in IQ, when in reality they do

39
Type I and Type II Errors
40
Type I and Type II Errors

• The probability of making a Type I error is equal
  to the confidence level of the statistical test
  (? 0.05 or ? 0.01)
• When you lower the probability of making a Type I
  error (e.g., use ? 0.01 instead of ? 0.05)
  you increase the probability of making a Type II
  error

41
Forget About It!

• For this class, you do not need to know how to
  determine the numerical value of a Type II error,
  nor do you need to understand power
• You do need to understand what a Type II error is

42
Consider a Sampling Distribution of Arts
Students GPAs.
Sampling error
?x ? x
6
10
43
What might this mean?

• This samples mean (10) appears to be
  substantially larger than the population mean
  (6). Why might this be?
• Perhaps there is something distinct about this
  sample such that it is not really part of this
  sampling distribution to begin with (e.g., maybe
  there are gifted arts students)
• Alternatively, perhaps its just fluke, and we
  just happened to have sampled a bunch of good
  arts students. Stated differently, perhaps this
  sample mean is part of the sampling distribution
  of arts students

44
Reminder

• We can determine the proportion of scores (in
  this situation, sample means) that would fall to
  the right of the sample mean in question by
  looking at a normal distribution table (Table
  E.10).
• To do so, we need to know the Z value of this
  sample mean. We will come back to this (but for
  sake of clarity, note that we will be learning to
  calculate a z-test, which uses a slightly
  different formula than the z-score formula that
  you know)

45
One vs. Two Tailed Tests

• The tails of a test set up our rejection region
  they determine how we decide to retain or
  reject Ho
• When we use a one-tailed test, we are testing the
  null hypothesis for a directional alternative
  hypothesis (e.g., Ha women will have higher IQs
  than men)
• We are only interested in whether or not women
  have higher IQs than men, not lower

46
Two-Tailed Tests

• When we use a two-tailed test, we are testing the
  null hypothesis for a non-directional alternative
  hypothesis (e.g., Ha women and men will have
  different IQs)
• Here, we are interested in whether or not women
  have higher or lower IQs than men

47
One vs. Two Tailed Tests (using ? 0.05)
2.5 2.5 5 5
48
Two-Tailed Tests

• Once we begin discussing t-tests, you will see
  that the value that determines whether or not our
  observed statistic falls above or below the ?
  0.05 depends on a number of factors
• For now, know that we reject Ho if our observed
  statistic is significantly greater than our
  expected statistic

49
Test Statistics

• A test statistic is a number calculated from the
  scores of a sample that allows us to test a Null
  Hypothesis and make a decision to reject or
  retain the Ho
• We will be talking about various test statistics
  for the remainder of the term, and will begin
  with the z-statistic today

50
Z-scores Revisited

• We know, by using the z-score formula, the
  probability of obtaining a score less than a
  given X value in a standard normal distribution
• E.g., when

51
The smaller portion area is .0668 (from Table
E.10)
.0668
z -1.5 0 X
70 100
52
Interpreting z

• This means that if we randomly select one score
  from this sample, the probability of that score
  being less than 70 is .0668
• But what if we want to test the hypothesis that a
  sample of n scores (mean 70) is actually a part
  of the population (mean 100, sd 20)?
• We no longer use the z-score formula, we use a
  z-statistic
• Remember whenever we are testing a hypothesis,
  we use a test statistic

53
What is Sigma?

• Usually, we do not know sigma ( ), the sd for
  a population (because obtaining data for an
  entire population is usually not done)
• Sometimes we do know sigma (e.g., for common
  psychological tests)
• When we know sigma, we can obtain the sampling
  distribution of the mean when the Null Hypothesis
  is true (that the sample does come from the
  population)

54
Null Hypothesis

• When we compare a sample mean with a population
  mean, the Null Hypothesis is that the sample DOES
  come from that population
• Ho or that 70 100
• But how can 70 100??
• Recall that a sample extracted from a population
  with µ 100 will more than likely result in a
  sample mean that is above or below 100 because of
  sampling error
• When we test a Null Hypothesis, we are testing to
  see if the sample mean and population mean are
  statistically different from each other (that
  there is a 95 chance based on an alpha level of
  .05 that 70 is statistically different from 100)

55
Sampling Distribution of the Mean

• In hypothesis testing, we set up the sampling
  distribution of the mean and then calculate a
  test statistic to determine if we can reject the
  Ho
• How is this done? Whenever we know we use a
  z-test we know for the one sample of
  interest, we know for the population, so we
  can calculate (standard error for the
  sampling distribution of the mean)

56
Standard Error Revisited

• Last week, we stated that the standard deviation
  of a sampling distribution of the mean is called
  standard error
• Standard error is used in test statistic formulae
  because we are using sampling distributions of
  the mean

57
Z Statistic

• If testing a null hypothesis that a sample mean
  is equal to the population mean (and sigma is
  known), we must use the following formula for the
  z-statistic (standard error instead of standard
  deviation)

58
The z-statistic

• Why zobs?? When we test the Ho, we will compare
  this zobs (our z observed) value with a zcrit
  (our z critical) value
• Note zobs is often also called zobt (for z
  obtained)
• Hypothesis testing compares the absolute value of
  zobs and zcrit in the following way
• If zobs gt zcrit we reject the null hypothesis
• If zobs lt zcrit, we retain the null hypothesis
• If zobs zcrit, we retain the null hypothesis

59
Zcrit

• The zcrit value is determined based on the alpha
  level used (usually alpha .05)
• zcrit is the z-score below which the probability
  that the sample data come from the population is
  less than .05 (the score that marks the tail)
• We use Table E.10 to determine zcrit
• Why might we be interested in this?
• We will know if we are using a one-tailed or
  two-tailed z-test based on our research question
• If we use a one-tailed test, the area in that
  tail is .05
• If we use a two-tailed test, the area in EACH
  tail is .025 (.05/2 tails)

60
Determining zcrit

• When we discussed z-scores, we reviewed problems
  where you know the proportion of scores and
  needed to determine the z-score (e.g., the
  lowest 10)
• Determining zcrit is a similar process
• Step 1 one-tailed or two-tailed?
• Step 2 alpha .05 or alpha .01?
• Step 3 Find the area in the smaller portion
  column in Table E.10 to determine the zcrit

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Tail Review
2.5 2.5 5 5
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zcrit for Two-tailed Tests

• Alpha .05 means that there is .025 per tail
• Find .025 in the smaller portion column
• zcrit 1.96
• Note! This is two-tailed, so this means
• zcrit 1.96
• Alpha .01 means that there is .005 per tail
• Find .005 in the smaller portion column
• zcrit 2.57
• Note! The exact smaller portion of .005 is not
  in the table. The values of .0049 and .0051 are
  listed, so which do we use?? Convention dictates
  that we use zcrit 2.57

63
zcrit for One-tailed Tests

• Alpha .05 means that there is .05 in the tail
• Find .05 in the smaller portion column zcrit
  1.64
• Note The exact smaller portion of .05 is not
  in the table. The values of .0495 and .0505 are
  listed, so which do we use?? Convention dictates
  that we use zcrit 1.64
• Note! To determine if this is a or zcrit,
  look at your Alternative Hypothesis
• Alpha .01 means that there is .01 in the tail
• Find .01 in the smaller portion column zcrit
  2.33
• Note! .0099 and .0102 are listed we use .0099 (z
  2.33) because it is closest to .0100
• Note! To determine if this is a or zcrit,
  look at your Alternative Hypothesis

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Z-test Hypothesis Testing Steps

• 1. State level of significance
• ? 0.05 (? 0.05 is usually used)
• OR ? 0.01
• 2. State IV, levels of IV, and DV
• 3. State the hypotheses
• Null hypothesis Ho
• Alternative Hypothesis Ha
• Note! At this point you need to read the question
  carefully to decide if you are testing a
  directional or nondirectional hypothesis

65
Z-test Hypothesis Testing Steps

• 4. Determine if you are using a one-tailed or
  two-tailed test
• A one-tailed test is used when you test a
  Directional hypothesis
• A two-tailed test is used when you test a
  nondirectional hypothesis
• 5. Find the rejection region
• I.e., find your zcrit!
• It is usually a good idea to draw the normal
  curve and plot your zcrit at this point it
  helps!

66
Z-test Hypothesis Testing Steps

• 6. Calculate your z statistic (zobs)
• 7. Compare zcrit to zobs
• Plot zobs on your normal distribution
• Compare the numerical value of zcrit to zobs

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Step 7 Example 1 (alpha .05)
.025
.025

• If zobs 2.59

Two-tailed
zcrit -1.96
1.96
zobs
2.59
.05
One-tailed
zcrit
1.64
zobs
2.59
68
Step 7 Example 2 (alpha .05)
.025
.025

• If zobs -1.75

Two-tailed
zcrit -1.96
1.96
zobs -1.75
.05
One-tailed
zcrit -1.64
zobs -1.75
69
Step 7

• Null Hypotheses are rejected when zobs falls in
  the rejection region (the area beyond zcrit).
  The rejection region is the tail of the
  distribution
• OR Null Hypotheses are rejected when
  zobs gt zcrit
• BUT! What about when the zobs and zcrit are both
  negative numbers??
• In this case, think of rejecting Ho when the
  absolute value of zobs gt zcrit
• Absolute value means that you remove the
  negative sign from both numbers (e.g., the
  absolute value of 5.5 is 5.5)

70
Step 8 (Last One)

• Step 8. State conclusions in words
• Once you decide to reject or retain Ho, you need
  to state your conclusions
• So what does rejecting the Ho actually mean for
  this research study?
• OR What does retaining the Ho actually mean for
  this research study?

71
Step 8 continued

• Rejecting the Ho for z-tests means that the
  sample mean is significantly different than the
  population mean, i.e., there is less than a 5
  chance that a sample extracted from this
  population would result in such a sample mean
  (because its in the tail end)
• BUT! For one-tailed tests, make sure that you
  state how they are different (i.e., is the sample
  mean greater or less than the population mean)
• Your conclusions should be clear enough that
  anyone in the general public could understand
  what the study found

72
An Example Using the Z-test

• Scientists have come up with a breakthrough new
  drug, they assert that by taking this drug it
  will affect your IQ. Because it is so new they
  are hoping it makes you smarter, but at this
  point it might also make you dumber. A sample of
  36 people has x 105, the population µ 100 and
  the population ? 15. Test their hypothesis.

73
Example cont.

• 1. State level of significance - ? 0.05 (what
  is usually used)
• 2. State IV and DV
• IV pill (levels pill and no pill)
• DV IQ scores
• 3. Null hypothesis
• The drug does not make you smarter or dumber
  (i.e., the sample mean does not differ from the
  population mean)
• Alternative Hypothesis
• The drug makes you either smarter or dumber

74
Example

• 4. B/c this hypothesis is non-directional, we use
  a two-tailed test
• 5. Find the rejection region ? 0.05, so with a
  two-tailed test we want a critical value that
  represents a region of rejection that makes up
  0.025 of the area of each tail

.025
.025
75
Example

• From Table E.10, we find that the critical value
  for z is equal to 1.96 or 1.96
• This means that zcrit ?1.96
• 6. Calculate your statistic

76
Example

• This means our zobs 2.00
• 7. Compare zcrit to zobs
• Is zobs gt zcrit??
• Yes! 2.00 gt 1.96

.025
z -1.96 1.96 2.00
77
Example

• B/c our zobs lies beyond zcrit we say our z-value
  falls into the region of rejection the value of
  zobs is greater than the value of zcrit so we
  choose to reject the Ho
• 8. We conclude that the IQ pill significantly
  changes someones IQ when they ingest it

78
Work On It

• The average number of times that a Canadian
  donates blood by the time they reach the age of
  50 is 10, with a population standard deviation of
  3 times. Researchers think that nurses donate
  more blood than average Canadians. 25
  fifty-year-old nurses are asked how many times
  they have given blood, and their mean number of
  times donating blood is 15. Test the hypothesis
  at the .01 level of significance.