Chapter 8 - Inferential Statistics, Hypothesis Testing and One Sample t-test

Middle Values in Normal Distribution

• 95%

• high probability values

• indicate that the treatment has no effect

  • same as pre-treatment expected levels

Extreme Values in Normal Distribution

• 5%

• scores that are very unlikely to be obtained from the original population

• provide evidence of a treatment effect

Hypothesis Test

• a statistical method that uses sample data to evaluate a hypothesis about a population

  • used in research to evaluate results

Null Hypothesis conclusions

• we either:

  • “reject the null hypothesis”

  • “fail to reject the null hypothesis”

H0

• null hypothesis symbol

Null Hypothesis

• the mean will not change because the treatment has no effect

H1

• alternative hypothesis

Alternative Hypothesis

• the mean will not be the same because something is going to happen/change

Alpha Level

• a probability value that is used to define the concept of “very unlikely”

  • it determined what the threshold is for “different enough”

α = .05

• 5% chance that the sample mean would be this extreme if the null were true

  • 95% of scores are likely values if null is true (because there is no efffect)

  • 5% of scores are unlikely values if null is true (because there is an effect so scores will be more extreme after treatment)

    • so we are never 100% sure of an effect — it’s always possible that there could just be an outliar in the extreme 5%

Errors in Hypothesis Testing

• there is a chance that we reject the null hypothesis when we shouldn’t have or we fail to reject the null hypothesis when we should have

  • type I errors and type II errors

Type I Errors

• reject the null hypothesis when it was actually true

  • reporting an effect that actually isn’t there

  • “false alarm”

    • maybe the sample just happened to be already in the extremes without the treatment — makes it look like there was an effect but there wasn’t

• want to keep risk of this error low

  • error = alpha level — alpha = .05 so error rate = .05

Type II Errors

• fail to reject the null hypothesis that is actually false

  • your data suggests no effect but there actually is one

  • “a miss”

  • your sample wasn’t in the critical region even though it should have been

  • denoted by the beta symbol: β

  • typically happens because your effect was too small (it moved the mean a little bit but not enough to get it into the critical region)

    • not enough power — small sample size, confounding variables, etc.

• we can’t determine the exact probability of this type of error

  • people are less concerned about this type

Statistically Significant

• the result is very unlikely to have occurred if the null hypothesis was true (if there was no effect); surpassed the threshold of “different enough”

Bidirectional (two-tailed) hypothesis test

• makes a prediction without indicating positive or negative

Directional (one-tailed) hypothesis testing

• predict the direction of your effect — then you can ONLY test that one direction

Directional (one-tailed) hypothesis testing — problems

• your prediction could be wrong

  • you might predict a negative effect but it turns out to be positive — you can’t test the positive side though so you would not see an effect

• easier to reach the critical region so there is more room for Type I errors

  • you need to strongly justify your use

When do we use t-scores instead of z-scores

• when we don’t know the population standard deviation

Estimated Standard Error

• an estimate of the real standard error when the population standard deviation is unknown

Degrees of Freedom (df)

• the number of scores in a sample that are independent and free to vary

• n-1

  • how many pieces of information do you need to find the mean

  • only need to know 2/3 (if sample is 3) because the last valve has to be a specific number to equal the mean

    • the final score is not free to vary (dependent on the other scores)

t-statistic

• comparing our mean to the null hypothesis to see if they are different enough to have an effect

3 types of t-tests

• one sample t-test

• independent samples t-test

• paired samples t-test

One sample t-test

• comparing 1 sample mean to 1 known population mean (but not std)

  • unique test but used all the time in psychology

Independent samples t-test

• comparing 2 means from separate groups

  • comparing 2 conditions (different people in each condition)

Paired samples t-test

• comparing 2 means from the same people

  • comparing 2 conditions (same people in both conditions)

One Sample t-test — Occurs when…

• you know a population mean and want to compare a mean to it

• you have a specific number you want to compare your sample mean to

Effect Size

• tells you magnitude of your effect and is not dependent on the sample size

d = 0.2

• small effect

d = 0.5

• medium effect

d = 0.8

• large effect

What boosts the likelihood of a significant effect

• lower error value (lower estimated standard error)

  • because we divide by estimated standard error, a larger error value will produce a smaller t-statistic

  • larger standard error → smaller t-statistic (not good!)

• larger sample size

  • because we divide by n to get standard error, a larger n value will produce a smaller standard error

  • as sample size increases, standard error decreases, and the likelihood of rejecting the null (and finding a significant effect) increases