hypothesis testing

Question 1

What are the types of hypotheses?

Answer 1

Research hypothesis (the question being investigated)

Null hypothesis (𝐻0): The hypothesis that is tested

Alternative hypothesis (𝐻1 or 𝐻𝐴): The opposite of 𝐻0

Question 2

What is a confidence interval?

Answer 2

An interval (lower & upper limit) within which the true value of a population parameter lies with a specified confidence level.

Question 3

How do hypothesis tests differ from confidence intervals?

Answer 3

Hypothesis tests assess whether a single value is the true parameter.

Confidence intervals estimate a range where the true parameter likely falls.

Question 4

What are the steps in a hypothesis test?

Answer 4

1. Specify the hypothesis
2. Obtain a test statistic from the data
3. Compare the test statistic to a reference distribution

Question 5

What is the null hypothesis (𝐻0)?

Answer 5

The hypothesis being tested, the test determines how much evidence the data provides to support this hypothesis.

Example:
𝐻0: 𝜇 = 3 (Mean density is 3 birds/km²)

Question 6

What is the alternative hypothesis (𝐻1 or 𝐻𝐴)?

Answer 6

The hypothesis that contradicts 𝐻0, suggesting a difference or effect.

Example:
𝐻1: 𝜇 ≠ 3 (Mean density is not 3 birds/km²)

Question 7

What are one-tailed and two-tailed tests?

Answer 7

One-tailed: Tests for a directional effect (e.g., 𝐻1: 𝜇 < 3 or 𝐻1: 𝜇 > 3)

when an effect can only occur in one direction

an effect can occur in both directions but only one direction is of interest.

Question 8

What is two-tailed tests?

Answer 8

Two-tailed: Tests for any difference (e.g., 𝐻1: 𝜇 ≠ 3)

Question 9

What do we compare the test statistic to?

Answer 9

A reference distribution (e.g., t-distribution) to determine if the observed difference is significant.

Question 10

How does variability affect hypothesis testing?

Answer 10

Low variability → Easier to detect a true difference

High variability → Harder to conclude significant differences

Question 11

What types of statistical tests are covered?

Answer 11

One-sample & two-sample t-tests

ANOVA (more than two groups)
z-tests (proportions)
Chi-square tests (categorical data)
Linear regression (t and F tests)

Question 12

How is a t-statistic calculated in a one-sample t-test?

Answer 12

tstat = (data estimate - hypothesised value) / SE(data estimate)
​

Question 13

what will the tstat be if H0 is true

Answer 13

the test statistic (𝑡𝑠𝑡𝑎𝑡
) will be small (dependent on sampling variability) because the difference between the data-estimate (sample mean) and the hypothesised value is small.

Question 14

what will the tstat be if H0 is false

Answer 14

the test statistic will be large (dependent on sampling variability) because the difference between the data-estimate and the hypothesised value is large.

Question 15

What distribution is typically used as the reference distribution in these examples?

Answer 15

The t-distribution is used as the reference distribution.

Question 16

What do the degrees of freedom (df) for the t-distribution depend on?

Answer 16

The degrees of freedom depend on what is being tested.

Question 17

What are the two ways the reference distribution helps determine the strength of evidence for the null hypothesis?

Answer 17

1. By obtaining an exact probability for the test statistic.
2. By comparing the test statistic to a critical value based on a predetermined significance level.

Question 18

In a one-sample two-tailed test, what is the null hypothesis (H0) and alternative hypothesis (H1)?

Answer 18

-𝐻0:𝜇=3.6

𝐻1:𝜇≠3.6

Question 19

What is the reference distribution for the test with
𝑛=16?

Answer 19

𝑡𝑑𝑓=𝑛−1=𝑡15

Question 20

In a two-tailed test, how is the area in the two tails interpreted?

Answer 20

The area in the tails represents the probability of obtaining a test statistic as extreme or more extreme than the observed value.

Question 21

How do you calculate the area in the two tails for the test statistic −0.753?

Answer 21

Add the area in the left tail (< -0.753) and the right tail (> 0.753): 0.226+0.226=0.452

Question 22

What is a p-value in hypothesis testing?

Answer 22

The p-value is the probability of observing a test statistic as extreme, or more extreme, than the one observed, assuming the null hypothesis is true.

Question 23

What does the p-value quantify in hypothesis testing?

Answer 23

The p-value quantifies the chance of observing the data (or something more extreme) if the null hypothesis (H0) is true.

Question 24

When is the null hypothesis typically rejected based on the p-value?

Answer 24

The null hypothesis (𝐻0) is usually rejected when the p-value is very small.

Question 25

What are some common threshold values (significance levels) for p-values?

Answer 25

0.10 → No evidence against 𝐻0
0.05 → Weak evidence against 𝐻0
0.01 → Some evidence against 𝐻0
0.001 → Strong evidence against 𝐻0

Question 26

What does a large p-value indicate?

Answer 26

The test statistic is likely under 𝐻0

We fail to reject 𝐻0

Question 27

What does a small p-value indicate?

Answer 27

The test statistic is unlikely under 𝐻0

We reject 𝐻0 in favor of 𝐻1

Question 28

What does it mean to “fail to reject the null hypothesis”?

Answer 28

It means that the p-value is large, indicating that the test statistic is likely to occur if the null hypothesis is true. Therefore, there is no strong evidence against 𝐻0

Question 29

What does it mean to “reject the null hypothesis”?

Answer 29

It means the p-value is small, indicating that the test statistic is very unlikely to occur if the null hypothesis is true. This provides evidence in favor of the alternative hypothesis (𝐻1)

Question 30

What is the relationship between a large p-value and the test statistic?

Answer 30

A large p-value suggests that the test statistic is likely to occur under the null hypothesis, providing no strong evidence to reject 𝐻0

Question 31

What is Fixed Level Significance Testing in hypothesis testing?

Answer 31

Fixed Level Significance Testing involves comparing the test statistic to a critical value based on a fixed significance level (e.g., 0.05, 0.1) to decide whether to reject the null hypothesis.

Question 32

What are statistical tables used for in significance testing?

Answer 32

Statistical tables provide critical values (quantiles) for different reference distributions (like the t-distribution) at various significance levels.

Question 33

What does the critical value represent in significance testing?

Answer 33

The critical value is the threshold beyond which the test statistic would be considered extreme enough to reject the null hypothesis.

Question 34

In a two-tailed test with a 5% significance level, how is the significance level distributed in the tails?

Answer 34

2.5% in the lower tail
2.5% in the upper tail

Question 35

What is a two-sample t-test used for?

Answer 35

A two-sample t-test is used to compare the means of two groups to test if there is a statistically significant difference between them.

Question 36

What are the null and alternative hypotheses in a two-sample t-test?

Answer 36

𝐻0:𝜇𝐴−𝜇𝐵=0 (no difference between means)

𝐻1:𝜇𝐴−𝜇𝐵≠0 (a difference exists between means)

Question 37

What formula is used to calculate the test statistic in a two-sample t-test?

Answer 37

tstat = (μ^A-μ^B)−0) / SE(μ^A−μ^B)

Question 38

What happens if the test statistic is less extreme than the critical value in a fixed-level test?

Answer 38

The null hypothesis is not rejected, indicating that the observed data does not provide strong evidence against the null hypothesis.

Question 39

Why do we divide the significance level equally between two tails in a two-tailed test?

Answer 39

Because we are testing for a difference in either direction (greater or smaller), so the probability of an extreme result must be shared between both tails.

Question 40

What is the conservative assumption made about variance in the two-sample t-test?

Answer 40

It is assumed that the variances are unequal, making the test more conservative.

Question 41

What is a paired t-test used for?

Answer 41

A paired t-test is used to compare two dependent groups where observations in one sample are paired with observations in the other sample (e.g., before and after treatment).

Question 42

What is the formula for the paired t-test statistic?

Answer 42

tstat = (μ^d−0)/ SE(μ^d)

Question 43

How is the standard error for the paired t-test calculated?

Answer 43

SE(μ^d)=SDd / sqrt(n)

μ^d = Mean of the differences

𝑆𝐷𝑑 = Standard deviation of the differences

𝑛 = Sample size

Question 44

What are the key assumptions for t-tests?

Answer 44

Independence of data within and between groups

Normal distribution of data (assess via histograms or Shapiro-Wilk test)

t-tests are robust to non-normal data if sample sizes are similar.

Question 45

What should you do if normality is unreasonable in a dataset?

Answer 45

Use non-parametric tests, such as the Mann-Whitney-Wilcoxon test.

Question 46

What does the Mann-Whitney-Wilcoxon test compare?

Answer 46

It compares the ranks of two groups to test if their distributions are the same or different.

Question 47

How is the Mann-Whitney U test statistic calculated?

Answer 47

U=W− (n(n+1))/2
​
W = Sum of the ranks
𝑛 = Sample size

Question 48

What does a large p-value in the Mann-Whitney test indicate?

Answer 48

A large p-value suggests no evidence of a difference between the two groups.

Question 49

How can you assess the practical significance of a result?

Answer 49

Present the effect size and confidence interval to provide context beyond just statistical significance.

Question 50

What are the key steps in hypothesis testing?

Answer 50

Specify null and alternative hypotheses

Calculate a test statistic

Compare to a reference distribution

Use data to determine the strength of evidence for the null hypothesis