Week 2: Ordinary least squares (OLS) and bivariate regression

Question 1

The 6 steps of hypothesis testing

Answer 1

1. Ensure that assumptions are met
2. Formulate hypotheses
3. Determine the critical area from the appropriate sampling distribution
4. Calculate the test statistic
5. Make decision
6. State conclusions

Question 2

Step 1: Ensure that assumptions are met

Answer 2

• Random samples
• Independt samples
• Interval-ratio level of measurement
• Sampling distribution is normally distributed

What is the probability that we would observe a particular sample statistic given this population? How unusual is this? Does this suggest the null hypothesis is false?

Question 3

Step 2: Formulate hypotheses

Answer 3

• Null is always no difference, no (positive/negative) effect etc.
• Alternative is a difference OR a particular difference (greater than, less than, etc.)

Question 4

Step 3: Determine the critical area from the appropriate sampling distribution

Answer 4

Decision rule: Reject H0 if the found Z-Score (Z*) is less than the critical z-value.

Question 5

Critical value of z

Answer 5

A critical value of z (Z-score) is used when the sampling distribution is normal, or close to normal. Z-scores are used when the population standard deviation is known or when you have larger sample sizes.

Question 6

How to find critical value

Answer 6

Step 1: Subtract the confidence level from 100% to find the α level: 100% – 90% = 10%.

Step 2: Convert Step 1 to a decimal: 10% = 0.10.

Step 3: Divide Step 2 by 2 (this is called “α/2”).
0.10 = 0.05. This is the area in each tail.

Step 4: Subtract Step 3 from 1 (because we want the area in the middle, not the area in the tail):
1 – 0.05 = .95.

Step 5: Look up the area from Step in the z-table. The area is at z=1.645. This is your critical value for a confidence level of 90%.

For a 90% confidence level (Two-tailed test), for a one-tailed test; step 3 can be skipped

Question 7

Step 4(1): Calculate the test statistic

Answer 7

• For every sample statistic, there is a formula for its test statistic (not so important)
• The test statistic allows us to make probability statements in terms of the “standard” distribution

Question 8

Step 4(2): Using P-values

Answer 8

Reject H0 if p < 0,05 (ignore critical value)

Question 9

P-value

Definition

Answer 9

The probability under the assumption of no effect or no difference (null hypothesis), of obtaining a result equal to or more extreme than what was actually observed.

P-value < 0.05 lower is generally considered statistically significant

The P stands for probability ands measures how likely it is that any observed difference between groups is due to chance

Question 10

Step 5 and Step 6: Decision and conclusion

Answer 10

• Decision: H0 can/cannot be rejected
• Conclusion: With 95% certainty… (this only counts if an alpha-level was chosen of 0.05)

Question 11

Regression equation

Answer 11

Y = a + bX + e
- a = intercept
- b = slope
- X = the amount of the independent variable used
- Y = the amount of the dependent variable used
- e = error term; the error in predictiong the value of Y, given the value of X

Minimizes squared distances between point and line

Question 12

Ordinary Least Squares Estimation

Answer 12

• Logic: Minimize the sum of the squared residuals
• Residual is the difference between the actual value of Y and the predicted value of Y

Question 13

Limits of the Ordinary Leat Squares regression

Answer 13

• Collinearity between X-variables leads to misinterpretation of the coefficients
• More observations than X-variables are required
• Only one Y-variable can be modeled