Statistical Inference

Question 1

What are two types of statistical inference?

Answer 1

Point estimation and interval estimation
Statistical inference is using estimated sample statistics and distributional properties of the estimator to make statements about unobserved population parameters

Question 2

How do you construct a confidence interval when the mean is unknown?

Answer 2

Use the estimator for the mean

Question 3

How do you construct a confidence interval when the variance is unknown?

Answer 3

Use the estimator for the variance or standard deviation (sigma hat) and replace the normal critical values with the values from Student’s t distribution using n - 1 degrees of freedom

Question 4

When is the t distribution used?

Answer 4

When the observations come from a normal distribution but the mean and variance are unknown

Question 5

What is the 95% confidence interval for a normal sample with unknown mean and variance?

Answer 5

[(mu hat) - t_n-1^0.025(sigma hat), (mu hat) + t_n-1^0.025(sigma hat)]
Divide sigma hat by root n when dealing with the distribution of the sample mean

Question 6

What is the Central Limit Theorem?

Answer 6

For sufficiently large samples (n >= 40), the distribution of the sample mean will be approximately normal with mean mu and variance sigma squared over n, meaning estimators for mu and sigma can be used to construct CIs for the mean

Question 7

What assumption must hold to construct confidence intervals for the variance?

Answer 7

Population distribution is normal

Question 8

What is the chi squared distribution?

Answer 8

Χ_n-1² ~ (n-1)S_n-1²/σ²

Question 9

What is the 1 - α level confidence interval for variance?

Answer 9

[(n-1)S_n-1²/Χ_{α/2, n-1}², (n-1)S_n-1²/Χ_{1 - α/2, n-1}²]

Question 10

What is a hypothesis test?

Answer 10

A test of whether a parameter takes a particular value based on sample statistics

Question 11

What is a statistical test?

Answer 11

A rule for whether to reject a certain assumption given the available data

Question 12

What are the null and alternative hypotheses?

Answer 12

The null hypothesis makes a specific assertion about a population parameter, the null is tested against the alternative hypothesis

Question 13

What is the test statistic?

Answer 13

The random variable which will determine the outcome of the test

Question 14

What are the acceptance and rejection regions?

Answer 14

When the test statistic falls in the acceptance region the null hypothesis is not rejected, when it falls in the rejection region the null is rejected
A and R are mutually exclusive and exhaustive

Question 15

What are Type I and Type II errors?

Answer 15

Type I: rejecting a true null hypothesis
Type II: not rejecting a false null hypothesis

Question 16

What is the significance level (or size) of a test?

Answer 16

α = P(Type I error) = P(reject H₀|H₀ true)

Question 17

What is the relationship between Type I and Type II errors?

Answer 17

There is a trade-off: designing a test to minimise one will increase the other
This is the same trade-off as size and powerm we cannot choose both but we want the smallest β for a given alpha

Question 18

What is β?

Answer 18

P(Type II error) = P(not rejecting null|null false)

Question 19

What is the power of a test?

Answer 19

The probability of rejecting a false null
1 - β = P(reject null|null false)

Question 20

How can size and power be represented on a diagram?

Answer 20

If the null hypothesis is that the sample comes from a standard normal distribution and the alternative hypothesis is that the mean is actually two, a vertical line can be drawn at the critical value, then the area under the alternative pdf before the critical line is β = 1 - power and the area under the null pdf after the critical line is the size

Question 21

How do you conduct a hypothesis test for the mean?

Answer 21

Compare the test statistic (mu hat - mu)/(standard error) to the critical value (usually the t value with n - 1 degrees of freedom, if n > 40 then the normal value)
Watch out for CLT, one and two tailed tests

Question 22

How do you conduct a hypothesis test for the variance?

Answer 22

Assume/know normality
Compare the test statistic (n-1)s²/(null variance) to the critical value from the chi squared distribution

Question 23

How do you conduct a hypothesis test for proportion?

Answer 23

Assume/know normality (np, nq > 5, p not very close to 0 or 1)
Compare test statistic (p hat - p₀)/sqrt(p₀(1 - p₀)/n) and compare to the critical value from the z distribution

Question 24

How do you construct a confidence interval for proportion?

Answer 24

For a (1 - α) CI of population proportion, use boundaries (p hat) +- z_α/2 * sqrt((p hat)(1 - p hat)/n)