Module 6, Probability and Introduction to Hypothesis Testing

Question 1

Hypothesis Testing (5 steps)

Answer 1

1. state the null and alternative hypotheses (H0 and H1)
2. choose the level of significance
3. find critical values
4. find test statistic
5. draw a conclusion

Question 2

Research Hypotheses

Answer 2

expression of relationship between variables in words (can expect to find based on past theory or research)

Question 3

Statistical Hypotheses

Answer 3

• focuses on numerical expressions of relationships between variables
• null and alternative hypotheses
• what we actually test when doing statistical hypotheses

Question 4

Null and Alternative Hypotheses

Answer 4

a set of hypotheses that present two mutually exclusive conclusions about data
- we make statistical hypotheses about population parameters and not sample statistics as we want to know if the effect exists in the population and not just in sample

null hypothesis (H0):
- proposes that the hypothesized change, difference, or relationship does not exist in the population
- example: null hypothesis would be the conclusion that playing at home does not increase chances of winning in the NHL

alternative hypothesis (H1):
- proposes that the hypothesized change, difference, or relationship does exist in the population
- example: does represent that conclusion that playing at home does increase our chances of winning in the NHL

Question 5

H0: μ = 6 (what type of hypothesis is this)

Answer 5

null hypothesis
- when researchers conduct hypothesis testing they are testing the null hypothesis and not the alternative

Question 6

H1: μ ≠ 6 (what type of hypothesis of this)

Answer 6

alternative hypothesis
- playing at home makes the probability of winning the game not 50% or .05
- the probability of winning a home game would not be equal to 6 out of the 12 games, thus we can reject the null hypothesis, if the value of the statistic is above or below 6
- the “does not equal” sign allow for the possibility that there could be a home team advantage (the home team wins more than 6 OR loses more than 6 game in the population)
- ≠ (two tailed test) < OR > (one tailed test)

Question 7

Choose the level of significance: Why care about probability in research?

Answer 7

• it is feasible to study samples and thus why researcher collect data from samples to represent the population
• research is typically conducted with samples
• estimation population parameters based on sample statistics (want to draw conclusions more broadly from population) - this process is called inferential statistics
• as data does not come from the entire population, we must evaluate data using probability
  ◦ evaluate sample data

Question 8

Sampling Error

Answer 8

• the difference between statistics calculated from a sample and those from the population
• samples are imperfect representations of the population

Question 9

What data would convince us that H0 is false?

Answer 9

• the calculated statistic value is not strong enough evidence
  ◦ recall the discussion of
  sampling error
• we look for evidence that has a low probability of occurring if H0 is true (to reject a null hypothesis and show there is a statistically significant effect or relationship)

Question 10

Alpha (α): defining “low probability”

Answer 10

• the probability of the statistic used to make a decision to reject H0
• conventionally, α = .05, or a 5% chance of a statistic occurring if H0 is true

Question 11

Find Critical Values

Answer 11

we need to identify values of statistics that are associated with low probability:
- how many games out of 12 would the home team have to win to convince us that there was a home team advantage or disadvantage taking into accountability the possibility of sampling error
- set our alternative hypothesis in a way that would allow for a home court advantage or disadvantage
- the number of wins on the x-axis and the probability of the home team winning on the y axis, thus the probability of the home team winning 6 games out of a random sample of 12 games is .2256 (22.56%)

Question 12

Find Critical Values: Region of Rejection

Answer 12

• represents the values of a statistic whose combined probability is low enough that we could reject H0
• represent that there is an effect or relationship within the population, which in our example would be a home team advantage or disadvantage

Question 13

Find Critical Values: Region of Non-rejection

Answer 13

• represents the values whose probability is not low enough to allow us to reject H0
• too great of a chance we could be rejecting the null hypothesis due to a sampling error

Question 14

Critical Value

Answer 14

value of a statistic that separates the two regions (rejection and non-rejection)

Question 15

Find Test Statistic

Answer 15

• the test statistic that is calculated depends on the statistical analysis required to answer the research question and type of data
  ◦ example: t, r, F, x2, z
• if the value of the statistic calculated for the sample lies beyond the critical values, reject the null hypothesis, otherwise, do not reject the null hypothesis
  ◦ implies that the probability
  of having found a value as
  big as we did was not due to
  chance but it was due to the
  actual fact
  ◦ if the value does not lay
  beyond the critical values
  than we do not reject the
  null hypothesis and thus we
  conclude that it is unlikely
  that effect exists in the actual
  hypothesis
  ◦ we found that the home
  team won 10 times out of
  the 12 games, 10 exceeds
  the critical value of 9 and
  thus a probability that we
  got this particular statistic by
  chance is less than 5% and
  thus our analysis would
  support the theory that
  there is a home team
  advantage

Question 16

p-Values

Answer 16

• p – values refer to the probability of obtaining a statistic by chance/sampling error
• p <.05 then null hypothesis has been rejected (not likely due to chance)
• p > .05 the null hypothesis has not been rejected (probability of the obtained statistic due to chance was more than 5%)

Question 17

“Proof”

Answer 17

• as hypothesis testing is based on probabilities it cannot provide “proof”
  ◦ proof implies absolute
  certainty (when we read
  quantitative research we can
  never say “proves” a
  phenomenon)
  ◦ it can provide support (or
  lack of support) for a
  hypothesis only

Question 18

Errors

Answer 18

it is possible that when we make a decision about null hypothesis, it could be incorrect:
type I error and type II error

Question 19

Type I Error

Answer 19

• occurs when we reject the H0, when it should have accepted
• this means that the researcher is saying that an affect exists in the population when in reality it does not, this can result in a communication of false information

Question 20

Type II error

Answer 20

• occurs when we accept H0, when it should have been rejected
• imagine we say there was no effect of a performance enhancing drug based on their data when in fact there is an effect in the population

Question 21

Sample Size and Alpha

Answer 21

sample size:
- larger samples make it more likely that we can reject H0
alpha:
- the smaller the value of α, the more difficult it is to reject H0

Question 22

Non-Directional Hypotheses

Answer 22

• do not specify the direction of the relationship (home team advantage or disadvantage)
  ◦ includes a ≠ in the numeric
  statement
  ◦ regions of rejection in both
  extremes of the distribution
  ◦ “two-tailed”

Question 23

Directional Hypotheses

Answer 23

• specify the direction of the relationship (in our example we might have stated the alternative hypothesis such that the region of rejection could only reject the null hypothesis if there was a home team advantage, not allowing for a home team disadvantage)
  ◦ includes a > or < in the
  numeric statement
  ◦ region of rejection in only
  one extreme of the
  distribution
  ◦ “one-tailed”