16 - Hypothesis Testing for 1 and 2 Samples

Question 1

When testing the mean of a normal distribution with unknown variance, what distribution do we use?

Answer 1

Student-t distribution

Question 2

What is a crucial assumption when using the student-t test?

Answer 2

The underlying data is normally distributed.

Question 3

Give the formula for the test statistic when comparing two means of independent normal distributions with known population variances. (Formula booklet)

Answer 3

(X̄ - Ȳ) - (μx - μy) / √ (σx²/nx + σy²/ny)

Question 4

What conditions must be met to use the z-test for the difference between two means?

Answer 4

Both samples are taken from normally distributed populations.
The variances are known for each distribution.
The two normal distributions are independent.

Question 5

What is a Pooled Estimate of Variance used for?

Answer 5

Comparing the means of two populations with small samples and/or unknown population standard deviations, assuming equal variances.

Question 6

Give the formula for the pooled estimate of variance (s_p^2). (Formula Booklet)

Answer 6

s_p^2 = [(nx - 1) sx^2 + (ny - 1) sy^2] / (nx + ny - 2)

Question 7

Give the formula for the test statistic (t) when using a pooled estimate of variance.
(Formula Booklet)

Answer 7

t = (X̄ - Ȳ) - (μx - μy) / [s_p * √ (1/nx + 1/ny)]

Question 8

What conditions must be met to use the t-test with pooled variance?

Answer 8

Both samples are taken from normally distributed populations.
The variances are equal and unknown.
The two normal distributions are independent.

Question 9

When testing the difference between two binomial proportions, what conditions must be met to use a normal approximation?

Answer 9

np > 10 and n (1 – p) > 10 for both samples.

Question 10

Give the formula for the test statistic when testing the difference between two binomial proportions.

Answer 10

(p1 - p2) / standard error, where standard error = square root p (1 - p) (1/n1 + 1/n2)] and p = (p1n1 + p2n2) / (n1 + n2)

Question 11

What is the null hypothesis when testing for a difference in population proportions (π) between two populations A and B?

Answer 11

H0: πA = πB

Question 12

How can you decrease the chance of a Type I error?

Answer 12

Lower the significance level (alpha).

Question 13

How can you decrease the chance of a Type II error?

Answer 13

Increase the sample size or increase the significance level (alpha).