Hypothesis Testing & t-Tests

Question 1

What is hypothesis testing?

Answer 1

It is used to determine whether a regression coefficient (𝛽𝑗) is statistically significant

Question 2

What are the 2 types of hypothesis?

Answer 2

H0 (null hypothesis): the coefficient is not significant (e.g 𝛽1
= 0)

H1 (alternative hypothesis): the coefficient is significant (e.g 𝛽1 doesn’t = 0)

Question 3

What is the general structure of hypothesis testing?

Answer 3

1) state the null and alternative hypothesis

2) compute the Test Statistic (e.g t-statistic)

3) compare to a critical value (from the statistical table)

4) make a decision:

• if t is greater than the critical value = reject H0

if t is smaller = fail to reject H0

Question 4

What is the T test?

Answer 4

Most commonly used test

used to check if an estimated coefficient (𝛽^𝑗) is significantly different from 0

Question 5

T test formula

Answer 5

t= β^j−β0 / SE(β^j)

​β^j = estimated coefficient

β0 = hypothesised value (usually 0)
​
SE(β^j) = standard error of β^j

Question 6

Give an example of a T-test

Answer 6

lets use the the following wage regression:

wage =β0 + β1 (education) + u

from the OLS we get:

β1 = 2.5, SE(β^1) = 0.8

We test:

H0 : β1 = 0 (education has no effect on wages)

H1 : β1 =/ 1 (does affect wages)

Now work out the T statstic:

t = (2.5-0)/ 0.8 = 3.125

Compare to the critical value

at the 5% significance level (2 tailed) it is roughly 2

as 3.123<2 we reject H0

Question 7

what is a two tailed t test used for?

Answer 7

Used when testing whether a coefficient is significantly different from 0, without specifying direction (increase or decrease). The significance level is split across both tails of the t-distribution.

Question 8

when is a one tailed test used?

Answer 8

Used when you are testing for a specific direction (increase or decrease but not both)

Question 9

what is a confidence interval?

Answer 9

it is a range of values that we expect the true coefficient (βj) to be with a certain level of confidence
​

Question 10

What is the formula for a confidence interval?

Answer 10

β^j​ +or - (t a/2 x SE(β^j​))

β^j​ = estimated confidence interval

SE(β^j​) = standard error of β^j​

ta/2 = critical value from the t-table (depends on confidence level)

Question 11

give an example of using confidence intervals

Answer 11

Let’s say we estimate the following regression model:

wage =𝛽0+𝛽1(education)+u

We get the estimated coefficient and standard error:

𝛽^1=2.5

𝑆𝐸(𝛽^1)=0.8

Using a 95% confidence interval (𝑡𝛼/2=2) we calculate:

2.5±(2×0.8)

2.5±1.6

[0.9,4.1]

This means that we are 95% confident that the true effect of education on wages is between 0.9 and 4.1.

Question 12

how can we use confidence intervals to test the hypothesis?

Answer 12

We check if zero is inside the confidence interval:

✅ If 0 is NOT in the CI (e.g., [0.9, 4.1]) → We reject
𝐻0, meaning 𝛽1 is statistically significant.

🚫 If 0 IS inside the CI (e.g., [-1.2, 3.8]) → We fail to reject
𝐻0, meaning there is not enough evidence that 𝛽1
is different from zero.