1 / 30

# Major Points - PowerPoint PPT Presentation

Major Points. Formal Tests of Mean Differences Review of Concepts: Means, Standard Deviations, Standard Errors, Type I errors New Concepts: One and Two Tailed Tests Significance of Differences . Important Concepts. Concepts critical to hypothesis testing Decision Type I error

## Related searches for Major Points

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

### Download Presentation

Major Points

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

### Major Points

• Formal Tests of Mean Differences

• Review of Concepts: Means, Standard Deviations, Standard Errors, Type I errors

• New Concepts: One and Two Tailed Tests

• Significance of Differences

### Important Concepts

• Concepts critical to hypothesis testing

• Decision

• Type I error

• Type II error

• Critical values

• One- and two-tailed tests

### Decisions

• When we test a hypothesis we draw a conclusion; either correct or incorrect.

• Type I error

• Reject the null hypothesis when it is actually correct.

• Type II error

• Retain the null hypothesis when it is actually false.

### Type I Errors

• Assume there are no differences (null hypothesis is true)

• Assume our results show that they are not same (we reject null hypothesis)

• This is a Type I error

• Probability set at alpha ()

•  usually at .05

• Therefore, probability of Type I error = .05

### Type II Errors

• Assume there are differences (alternative hypothesis is true)

• Assume that we conclude they are the same (we accept null hypothesis)

• This is also an error

• Probability denoted beta ()

• We can’t set beta easily.

• We’ll talk about this issue later.

• Power = (1 - ) = probability of correctly rejecting false null hypothesis.

### Critical Values

• These represent the point at which we decide to reject null hypothesis.

• e.g. We might decide to reject null when (p|null) < .05.

• Our test statistic has some value with p = .05

• We reject when we exceed that value.

• That value is the critical value.

### One- and Two-Tailed Tests

• Two-tailed test rejects null when obtained value too extreme in either direction

• Decide on this before collecting data.

• One-tailed test rejects null if obtained value is too low (or too high)

• We only set aside one direction for rejection.

### One- & Two-Tailed Example

• One-tailed test

• Reject null if number of red in Halloween candies is higher

• Two-tailed test

• Reject null if number of red in Halloween candies is different (whether higher or lower)

### Within subjects t tests

• Related samples

• Difference scores

• t tests on difference scores

• Advantages and disadvantages

### Related Samples

• The same participant / thing give us data on two measures

• e. g. Before and After treatment

• Usability problems before training on PP and after training

• Darts and Pros during same time period

• With related samples, someone high on one measure probably high on other(individual variability).

Cont.

### Related Samples--cont.

• Correlation between before and after scores

• Causes a change in the statistic we can use

• Sometimes called matched samples or repeated measures

### Difference Scores

• Calculate difference between first and second score

• e. g. Difference = Before - After

• Base subsequent analysis on difference scores

• Ignoring Before and After data

### Results

• Pros got more gains than darts

• Was this enough of a change to be significant?

• If no difference, mean of computed differences should be zero

• So, test the obtained mean of difference scores against m = 0.

• Use same test as in one sample test

### t test

D and sD = mean and standard deviation of differences.

df = 100 - 1 = 9 - 1 = 8

Cont.

### t test--cont.

• With 99 df, t.01 = +2.62 (Table E.6)

• We calculated t = 2.64

• Since 6.64 > 2.62, reject H0

• Conclude that the Pros did get significantly more than Darts

### Advantages of Related Samples

• Eliminate subject-to-subject variability

• Control for extraneous variables

• Need fewer subjects

### Disadvantages of Related Samples

• Order effects

• Carry-over effects

• Subjects no longer naïve

• Change may just be a function of time

• Sometimes not logically possible

### Between subjects t test

• Distribution of differences between means

• Heterogeneity of Variance

• Nonnormality

### Pros during ups and downs in DOW

• Effect of fluctuations in DOW: did it effect Pros

• Different question than previously

• Now we have two independent groups of data

• Pros during positive DOW, and Pros during negative DOW

• We want to compare means of two groups

### Differences from within subjects test

Cannot compute pairwise differences, since we cannot compare two random data points

We want to test differences between the two sample means (not between a sample and population)

### Analysis

• How are sample means distributed if H0 is true?

• Need sampling distribution of differences between means

• Same idea as before, except statistic is (X1 - X2) (mean 1 – mean2)

### Sampling Distribution of Mean Differences

• Mean of sampling distribution = m1 - m2

• Standard deviation of sampling distribution (standard error of mean differences) =

Cont.

### Sampling Distribution--cont.

• Distribution approaches normal as n increases.

• Later we will modify this to “pool” variances.

### Analysis--cont.

• Same basic formula as before, but with accommodation to 2 groups.

• Note parallels with earlier t

### Degrees of Freedom

• Each group has 5 data points.

• Each group has n - 1 = 50 - 1 = 8 df

• Total df = n1- 1 + n2 - 1 = n1 + n2 - 2 50 + 50 - 2 = 98 df

• t.01(98) = +2.62 (approx.)

### Assumptions

• Two major assumptions

• Both groups are sampled from populations with the same variance

• “homogeneity of variance”

• Both groups are sampled from normal populations

• Assumption of normality

• Frequently violated with little harm.

### Heterogeneous Variances

• Refers to case of unequal population variances.

• We don’t pool the sample variances.

• We adjust df and look t up in tables for adjusted df.

• Minimum df = smaller n - 1.

• Most software calculates optimal df.