Topic 10: Tests for Mean

Question 1

Describe the Z test (1 sample)

Answer 1

Similar to Proportion test, BUT
- The population SD is known (in peanut data example, popSD was calculated)
- Usually continuous data, only applicable to binary data if the sample size is large enough

Question 2

Why do we usually prefer T test over Z test?

Answer 2

In Z test, the population SD is assumed to be known. In most cases, we only have the sample, not the population so the popSD cannot be known. In T test, popSD is replaced by sample SD.

Question 3

What is degree of freedom?

Answer 3

Degree of freedom is the degree of wide-ness of the TS distribution when the sample SD vary depending on the sample size

Question 4

State the TS formula for 1 sample T test

Answer 4

TS = (OV-EV)/(sampleSD/sqrt(n))

Question 5

Describe the paired T test

Answer 5

Answers to the question of “Is there any difference?”
–> H0: mean exhaustion time difference = 0

Both data sets will be used to generate the difference between them –> the differences is now 1 sample that we use

Question 6

Contrast SD and SE

Answer 6

• SD: how far each cyclist varies from the mean in this example sample of 9
• SE: how far the sample means vary from the true population mean (all elite cyclists)

Question 7

Describe hypothesis and assumptions of the 2 sample T test

Answer 7

• Hypothesis: H0 can be something like there is no change/no difference (for the question does red bull change the heart rate in students?) mean1=mean2
• Assumptions:
  + 2 samples are independent
  + 2 populations have equal spread (SD/variance)
  + 2 populations are Normal

Question 8

State the formula for TS in 2 sample T test

Answer 8

TS = (OV-EV)/SE = (mean1-mean2-0)/SE

SE = sqrt(poolSD^2 * (1/n1 + 1/n2))

poolSD^2 = (((n1-1)SD1^2) + ((n2-1)SD2^2)) / (n1+n2-2)

Question 9

Identify and describe the test used for equal spread + R code

Answer 9

Levene’s Test (F-Test) - tests the null hypothesis that the spread is equal between two populations

var.test()

Question 10

Identify and describe the tests used for normality

Answer 10

• Q-Q plot: graphs theoretical quantiles based on the normal curve vs. actual quantiles
  + If the dots can be connected by a fairly straight line –> conclude Normality
• Shapiro-Wilk test: tests the null hypothesis that the data is Normal
  + shapiro.test()

Question 11

How can comparative boxplots test normality and equality of variance?

Answer 11

• Normality: whether the boxes look symmetrical
• Equality of variance: how data is spread (the box and the line)

Question 12

Identify and describe situations in which assumptions for 2 sample T test are not met and other tests to replace

Answer 12

• 2 samples have unequal spread/SD/variance: Welch 2 Sample T test
  + t.test(data1,data2, var.equal=F)
• 2 samples are not Normal:
  + transformations or non-parametric tests (eg wilcox.test)
• 2 sample are not independent:
  + paired T-test (sample of difference)