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1
Q

Degrees of Freedom?

A
  • n - 1
  • Example, You have n numbers that must sum to 10
    • x1 + x2 + x3 + x4 = 10
    • First choice: you choose x1 = 13
    • Then, x2 + x3 + x4 = -3
    • Second choice: you choose x2 = 8
    • Then, x3 + x4 = -11
    • Third choice: you choose x3 = 3
    • Then, x4 = -14 (forced, no choice can be made)
    • You were able to make 3 choices before the 4th decision was forced, or you were able to make n-1 choices
2
Q

SEM

A
  • Standard Error of the Mean
  • S / sqrt(n) ; where S = sample standard deviation
3
Q

T-statistic?

A
  • t = (xbar - M) / SEM
  • the T-statisitic is used in a T-test when you are deciding if you should support or reject the null hypothesis
  • it is very similar to the z-score
  • you should use the T-statistic when you have a small sample size (less than 30), or if you don’t know the population standard deviation
  • the greater the T value, the more there is evidence your score is different from the average
  • a smaller T value is evidence that the score is NOT significantly different from the average.
4
Q

ME

A
  • Margin of Error
  • ME = tcritical * SEM
5
Q

CI

A
  • Confidence Interval
  • CI = xbar +/- ME
6
Q

Cohen’s d

A
  • standarized mean difference that measures the distance between means in standardized units
  • What is mean difference? (xbar - M)
  • What is S?
    • Standard deviation of the sample
  • Cohen’s d = xbar - M / S
7
Q

r2

A
  • coefficient of determination
  • 0.00 to 1.00
  • Value of ‘0’ means that the variables are not at all related
  • Value of ‘1’ means that the variables are perfectly related
    • This rarely happens
  • df = degrees of freedom
  • ​r2 = t2/t2 + df
  • NOTE: ‘t’ is not tcritical ; it is the t-value you get from the t-test
8
Q

Dgrees of freedom for a 2-dimensional array?

A
  • (n - 1)2
  • Example for a 4x4 array, you can make (4-1)2 choices, or 9 choices
9
Q

What is the major difference between the T-score and the Z-score?

A
  • The major difference between using a Z-score and T-score is that you have to estimate the population standard deviation with a T-score
  • The T-test is also used if you have a small sample size (less than 30)
10
Q

What is the t-critical value for a one-tailed alpha levle of 0.05 with 12 degrees of freedom?

A
11
Q

You have a sample of size 30. What are the t-critical values for a two-tailed test with a = 0.05?

A
  • t-table
  • Since it is a two-tailed test, you want to look for .025 with degrees of freedom of (n - 1) or 29
  • 2.045
12
Q

Your sample is size 24 and you get a t-statistic of 2.45.

The area to the right of the t-statistic is between __________ and _________ ?

A
  • (n - 1) = 23
  • t-table
  • Consulting the table for 23 degrees of freedom, 2.45 is between the t-statistics of 2.177 and 2.500
  • The t-statisitc of 2.177 aligns with a p-value of 0.02
  • The t-statistic of 2.500 aligns with a p-value of 0.01
  • So, the area to the right of the t-statisitc is between 0.02 and 0.01
13
Q

If the t-statisic is “far from” the popultaion mean, what do you do?

A
  • reject the null
14
Q

The LARGER or SMALLER the value of xbar, the stronger the evidence that M > Mo

A
  • Larger
15
Q

The LARGER or SMALLER the value of xbar, the stronger the evidence that M < Mo

A
  • Smaller
16
Q

The further the value of xbar from Mo in either direction, the STRONGER or WEAKER the evidence that M != Mo

A
  • Stronger
17
Q

What is the equation for a One-Sample t-test?

A
  • t = xbar - Mo / (s/sqrt(n))
18
Q

Givent the following parameters, calculate the t-statistic:

  • M = 6.07
  • n = 500
  • xbar = 6.47
  • S = 0.40

Should we reject Ho or Fail to reject Ho based on the t-statistic?

A
  • t = (6.47 - 6.07) / (0.40/sqrt(500))
  • t = .40 / (.40/22.36)
  • t = .40 / .1788854
  • t = 22.36
  • This t-statistic is very large, and therefore we should Reject Ho
19
Q

What is the P-value?

A
  • the P-value is probability above or below the t-statistic
20
Q

Given the following sample set and parameters, what is the t-statistic and ?

  • 5
  • 19
  • 11
  • 23
  • 12
  • 7
  • 3
  • 21
  • M = 10
  • a = 0.05
A
  1. Calculate S:
    1. Calculate the xbar (average of samples) = 12.625
    2. Calculate the squared variance:
      1. 58.14063
      2. 40.64063
      3. 2.640625
      4. 107.6406
      5. 0.390625
      6. 31.64063
      7. 92.64063
      8. 70.14063
    3. Sum the squared variance = 403.875
    4. Divide by (n - 1) or 7 = 57.69643
    5. Take the sqrt = 7.595817
  2. Calculate t:
    1. (xbar - 10) / (s/sqrt(n)
    2. (12.625 - 10) / (7.596 / sqrt(8))
    3. 2.625 / 2.68559155
    4. 0.977
21
Q

For a t-statistic of 0.977, calculate the P-value and what this means for a two tailed test with a = .05

A
  • GraphPad
  • .3611
  • With a two tailed test and alpha of 0.05, the upper and lower critical regions have an alpha value of .025
  • Since the p-value of .3611 is greater than .025, the p-value is not statistically significant and thus we fail to reject the null
22
Q

What are the critical values for a two-tailed test with a = 0.05 with n = 25

A
  • t-chart
  • Since a = 0.05, and it is a two-tailed test, we need to divide the value in 2, so our t-critical values are +/- 0.025
  • Since n = 25, we are interested in (n - 1) or 24
  • Consulting the chart for df = 24 and t-critical value of 0.025, the p-value is 2.064
23
Q

Given the following parameters, calculate the t-statistic:

  • M = $1700
  • S = $200
  • Mo = $1830
  • n = 25
A
  • t = (M - Mo) / (s/sqrt(n))
  • t = (1700 - 1830) / (200/sqrt(25))
  • t = -130 / 40
  • t = -3.25
24
Q

Calculate the confidence interval given the following parameters:

  • M = $1700
  • t-statistic = 2.064
  • S = $200
  • n = 25
A
  • 1 standard error (SE) = S / sqrt(n)
  • confidence interval (CI) = ( M - t(SE) , M + t(SE) )
  • CI = ( 1700 - 2.064( 200 / sqrt(25) ) , 1700 + 2.064( 200 / sqrt(25) )
  • CI = ( 1700 - 82.56 , 1700 + 82.56 )
  • CI = ( 1617.44 , 1782.56 )
25
Q

What is the equation for Margin of Error (ME)?

A
  • t * ( s/sqrt(n) )
26
Q

If n = 100 and S = 200, the margin of error is _______ ? (95% CI)

A
  • t-table
  • t = 1.984
  • ME = t * s/sqrt(n)
  • ME = 1.984 * (200 / sqrt(100))
  • ME = 1.984 * 20
  • ME = 39.68
27
Q

What are the two types of Effect Size measures?

A
  1. Difference Measures
    1. mean difference (easiest)
    2. standardized difference
  2. Correlation Measures
28
Q

What is the type of standardized difference used in this course?

A

Cohen’s d

29
Q

What is the Correlation Measure used in this course?

A
  • r2
  • proportin (%) of variation in one variable that is related to (“explained by”) another variable
30
Q

Statistical Significance

A
  • rejected the null hypothesis
  • results are not likely due to chance (sampling error)
31
Q

How do we know if resutls are meaningful?

A
  1. What was measured?
    1. Did the variables have practical, social, or theoretical importance?
  2. Effect size?
    1. How large were the results?
    2. This does not necessarily mean that small results are not important
  3. Can we rule out random chance?
    1. This doesn’t gaurentee that our results were important, but goes a long way in helping
  4. Can we rule out alternative explanations?
    1. lurking variables
32
Q

If t = 2, df = 20, then compute r2

A
  • t = 2
  • df = 20
  • ​r2 = t2/t2 + df
  • r2 = (2)2/(2)2 + 20
  • r2 = 4/24
  • r2 = .17 or 17%
33
Q

If t = 4, df = 10, the compute r2

A
  • t = 4
  • df = 10
  • ​r2 = t2/t2 + df
  • r2 = (4)2/(4)2 + 10
  • r2 = 16/26
  • r2 = .62 or 62%
34
Q

What goes into Results Sections?

A
  1. Descriptive Statistics (M, SD)
    1. Heart of our study and reports what happened
    2. Can be in text, graphs, or tables
  2. Inferential Statistics
    1. Hypothesis or confidence control (sometimes both)
    2. Need to tell the reader:
      1. What kind of test did you conduct? (e.g. one-sample t-test)
      2. Test statistic (the value of T)
      3. DF (degrees of freedom of the test)
      4. P-value
      5. Direction of the test (one-tailed or two-tailed)
      6. Alpha level
    3. APA Style
      1. t(df) = x.xx, p = .xx, direction
      2. t(24) = -2.50, p = .01, one-tailed
    4. Confidence Intervals
      1. Confidence Level, e.g. 95%
      2. Lower Limit
      3. Upper Limit
      4. CI on what? What was it for? What is it telling us?
        1. e.g. single mean or difference between two means
      5. Example of this:
        1. “Confidence on the mean difference; 95% CI = (4,6)
  3. Effec size measures
    1. Could be Cohen’s d, or r2
    2. APA Style:
      1. d = x.xx
      2. r2 = .xx (no leading zero because it cannot be greater than 1)
35
Q

Full One-Sample Test Example

  • “US families spent an average of $151 per week on food in 2012”
    • M = 151
  • A discount food store wants to implement a cost saving program.
    • Ho: the program did not reduce the cost of food
    • Ha: the program did reduce the cost of food
  • Statistical Symbols:
    • null -> Ho: Mprogram >= 151
    • alt -> Ha: Mprogram < 151
  • n = 25

Questions:

  1. What type of test will this be?
  2. Find t-critical if a = .05
  3. If S = $50, computer SEM; what does this value mean?
  4. If xbar = 126, what is the mean difference?
  5. Compute the t-statistic
  6. Does t fall within the critical region?
  7. Compute Cohen’s d
  8. Compute r2
  9. Compute the margin of error for 95% CI (2-tailed, df = 24, a = .05)
  10. Compute the 95% CI for the mean (lower limit, upper limit)
A
  1. one-tailed test in the negative direction
  2. t-critical = -1.711
  3. SEM = s / sqrt(n) = 50/sqrt(25) = 10
    1. This tells us that we expect sample means to differ from the true population mean by $10
    2. Most sample means should fall between $141 and $161
  4. Mean difference = xbar - M = 126 - 151 = -25
  5. t-statistic = (xbar - M) / SEM = -25/10 = -2.5
  6. Yes; the t-statistic (-2.5) falls below (-1.711)
  7. Cohen’s d = (xbar - M) / S = -25 / 50 = -0.50
  8. r2 = t2 / t2 + df = (-2.5)2 / (-2.5)2 + 24 = 6.25 / 30.25 = .21 or 21%
  9. margin of error = t-critical * SEM = 2.064 * 10 = 20.64
  10. 95% CI = xbar +/- margin of error = 126 - 20.64, 126 + 20.64 = (105.36, 146.64)
36
Q

What is the standard error for two normally distributed samples?

A

sqrt( (s12/n1) + (s22/n<span>2</span>) )

37
Q

What is the degrees of freedom for two normally distributed samples?

A

df = (n1 - 1) + (n2 - 1) = n1 + n2 - 2

38
Q

You and your friends want to go out to eat, but you don’t want to pay a lot. You decide to either go to Gettysburg or Wilma. You look online and find the average meal prices at 18 restaurants in Gettysburg and 14 restaurants in Wilma.

What do we need to know to compare these samples?

A
  • The sample averages
  • The size of each sample
  • The sample standard deviations
39
Q

Give the following parameters, calcuate the

  1. standard error
  2. t-statistic
  3. t-critical values
  4. What is the outcome?
  • xbarG = 8.94
  • xbarw = 11.14
  • SG = 2.65
  • SW = 2.18
  • nG = 18
  • nW = 14
  • DF = 18 + 14 - 2 = 30
A
  • SE = SxbarG - xbarW = sqrt( (SG2/nG) + (SW2/nW) )
  • SE = sqrt( (2.652/18) + (2.182/14) )
  • SE = .85
  • t-statistic = (xbarG - xbarW / SxbarG - xbarW) OR (xbarW - xbarG / SxbarG - xbarW)
  • t-statistic = (8.94 - 11.14) / .85 OR (11.14 - 8.94) / .85
  • t-statistic = -2.58 OR 2.58 = +/- 2.58
  • t-critical values = +/- 2.042
  • Outcome = we reject the null since 2.042 is below 2.58; prices between W and G are statistically different at a significance level of .05
40
Q

Given the following parameters for Drug A and Drug B, find the following:

  • xbarA
  • xbarB
  • SA
  • SB
  • Calculate the t-statistic
  • Calculate the t-critical values (a = .05; two-tailed)
  • Accept or reject the null?
    • Ho : M<span>A</span> = MB
    • HA : MA != MB
A
  • xbarA: 33.5%
  • xbarB: 31.2%
  • SA: 8.89% (sum of the square of variance for each sample)
  • SB: 10.16% (sum of the square of variance for each sample)
  • t-statistic:
    • t = xbarA - xbarB / sqrt( (SA2/nA )+ (SB2/nB) )
    • t = 33.5 - 31.2 / sqrt( (13.17) + (20.65) )
    • t = 2.3 / sqrt( 33.82 )
    • t = 2.3 / 5.82
    • t = .40
  • t-critical values:
    • two-tailed test; a = .05;
    • df = (6 +5 - 2) = 9
    • consulting the t-chart @ .025 and 9 = +/- 2.262
  • Accept or reject the null/
    • Since the t value of .40 is below the t-critical value of 2.262, we fail to rejct the null
41
Q

Given the following parameters, answer the questions:

  • xbar = 3.8
  • ybar = 2.1
  • nx = 18
  • ny = 25
  • Sp2 = 0.13
  • Ho: Mx - My <= 0
  • H<span>A</span>: Mx - My​ > 0
  • t-statistic = ______
  • t-critical = _______
  • Retain or Reject Ho?
A
  • t-statistic:
    • t = xbarA - xbarB / sqrt( (SA2/nA )+ (SB2/nB) )
    • t = 3.8 - 2.1/ sqrt( .13/18 + .13/25 )
    • t = 1.7/ sqrt( .007 + .005 )
    • t = 1.7/.11
    • t = 15.45
  • t-critical values:
    • one-tailed test; a = .05;
    • df = (18 + 25 - 2) = 41
    • consulting the t-chart @ .05 and 41 = +/- 1.684
      *