Topic 10 - z-tests & t-tests

Question 1

L.O.

Answer 1

LO7 [capstone] Given real multivariate data and a problem, formulate an appropriate hypothesis and perform a range of hypothesis tests.
LO8 Interpret the p-value, conscious of the pitfalls associated with testing.

Question 2

The z-test

Answer 2

• 1 sided or 2 sided
• HATPC process
• Used for mean or poportion

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Question 3

The t-test

Answer 3

For when we don’t know the popSD
- Uses sample SD and t-distribution
- Used to investigate differences in means
- t-distribution varies in shape according to DoF (n-1)

Question 4

When we don’t know the popSD

Answer 4

1. Estimate popSD from sampleSD and use z-test.
   - Adds extra variability to test statistic due to samples containing different SD
2. Using t-test

Question 5

t-Distribution

Answer 5

• Changes shape with DoF
• Bell shaped and symetrical
• 1DoF = lower peak and higher tails
• 20DoF = High peak and small tails

DoF = n-1
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Question 6

Comparing z-test and t-test

Answer 6

Z test:
- P-curve = normal curve
- Rcode = pnorm

T test:
- P-curve = tn-1
- Rcode = pt

BOTH:
test stat = (OV-popMean) ÷ (sampleSD/ root(n))

Question 7

Paired t-tests

HAT

Answer 7

H0: Mean for differences = 0
H1: mean for differences ≠ 0

Assumptions:
- Assume the population of differences is normal
- Each pair is independent

Test-stat:
- With a mean of 11.7 and sample SD of 10.8, we can substitute them into the formula to obtain the test stat;

11.7-0 ÷ (10.8 / root(9))
12.= 3.25

Question 8

Two-sample t-tests

Answer 8

2 groups of people

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Question 9

Assumptions for two-sample t-tests

Answer 9

• The 2 samples are independent (check context)
• The populations have equal spread (SD÷variance)
• The two populations are normal (check boxplots, no outliers, histograms, qqplot, normality test)

Question 10

QQplot for normality

Answer 10

Check if data is normally spread
- Graphs the theoretical quantities based on normal curve against the actual quantities
- If the line formed by the points is reasonably straight, then the data is normally distributed

Question 11

Shapiro-Wilk test for normality

Answer 11

• Tests the H0 that the data is normal:
  H0 = data is consistent with normal distribution
  H1 = Data is NOT consistent…

Test is very sensitive to sample size;
- small = normal
- large = not normal

Question 12

Levene’s test (F-test) for equal spread

Answer 12

For Two-sample t-test
Tests the H0 that the 2 populations have equal spread

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