Mod 9

Question 1

Recognize whether two samples are independent or paired

What are paired data?

Answer 1

from the same individual (matched pairs or repeated measure studies)

or from two units who form a natural pair (twins, couple, or matched subjects)

Question 2

Notation for paired data

What is:
- μd
- 𝞼d
- d-bar
- sd

Answer 2

μd = mean for population of differences

𝞼d = SD for population of differences

d-bar = mean for the sample of differences

sd = SD for sample of differences

Question 3

When are samples independent?

Answer 3

Samples are independent if the selection of the individuals who make up one sample does not influence the selection of the individuals in the other sample
- random sample obtained from each of two different populations

Question 4

Notation for independent samples

What is:
- μ1 - μ2
- 𝞼1 or 2
- x-bar1 - x-bar2
- s1 or 2

Answer 4

• μ1 - μ2 = difference in population means
• 𝞼1 or 2 = SD of population
• x-bar1 - x-bar2 = difference in sample means
• s1 or 2 = SD of sample (from first or second population)

Question 5

What is the sampling distribution of paired data (d-bar)?

Answer 5

d-bar ~ AN (μd, 𝞼/square root n or sd/square root n)

Question 6

When is the sampling distribution of paired data approximately normal?

Answer 6

• population of differences is bell-shaped and random sample is obtained
• The population of differences is not bell-shaped but a large random sample (n ≥ 30)

Question 7

Formula for confidence interval for paired data

What are the conditions for this formula?

Answer 7

d-bar +/- t* (sd/square root n)

t* comes from distribution with d.f. (n-1)

• shape of data distribution of differences is normal and random sample is obtained
• if nonnormal, n ≥30

Question 8

How to compute a test statistic for paired data?

What distribution does this test statistic come from?

Answer 8

d-bar - μdo/sd/square root n

comes from t-distribution with d.f. (n-1)

Question 9

How to find p-value for paired data?

Answer 9

t.dist (t, d.f. (n-1), 1)

Question 10

Conduct a paired data t-test (hypothesis testing for paired data)

Answer 10

1) Hypothesis
- Ho: μd = μdo
- Ha: μd < μdo, μd > μdo, μd = μdo

2) Conditions
- population of differences is normal and random sample is used
- if nonnormal, n ≥30

3) Compute test statistic
t = d-bar - μdo/sd/square root n

4) Compute p-value
t.dist (t, d.f. (n-1), 1)

5) Conclusion
- p-value ≤ alpha = RTN
- p-value ≥ alpha = FTRN

Question 11

What is the sampling distribution of independent samples (x-bar1 - x-bar2)?

Answer 11

x-bar1 - x-bar2 ~ AN (μ1 - μ2, square root of 𝞼1^2/n1 + 𝞼2^2/n2

Question 12

When is the sampling distribution of independent samples approximately normal?

Answer 12

• population of measurements are bell-shaped and random sample is used
• if nonnormal then n ≥ 30 from each population

Question 13

Formula for confidence interval for independent data

What distribution does the critical value come from?

Answer 13

x-bar1 - x-bar 2 +/- t* (square root of sd1^2/n1 + sd2^1/n2)

t-distribution with d.f. = min(n1-1, n2-1)

Question 14

How to compute test statistic for independent data (two t-test)?

Answer 14

t = (x-bar1 - x-bar2)/√s1^2/n1 + s2^2/n2

from t distribution with min (n1-1, n2-1) df

Question 15

How to compute p-value for independent data (two t-test)?

Answer 15

t.dist (x, df, 1) or tails from t.dist

Question 16

Conduct two-sample t test for a difference in population means (hypothesis testing for independent samples)

Answer 16

1) Hypotheses
H0: μ1 - u2 = 0, μ1 = μ2
Ha: μ1 - μ2 <,>, ≠ 0 or μ1 <, >, ≠ μ2

2) Verify conditions
The populations of measurements are both bell-shaped and random samples of any size are measured
Large random samples (both samples are at least 30) are measured from each population.

2) Compute test statistic
t = (x-bar1 - x-bar2)/√s1^2/n1 + s2^2/n2
t distribution with min (n1-1, n2-1) df

3) Compute p-value
t.dist (x, df, 1)

4) Conclusions
Statistically significant if p-value ≤ alpha
Not statistically significant if p-value ≥ alpha