Exam 2 Review

Question 1

13. Provide the formula for Bayes Rule in the case that event 𝐴 depends on event 𝐵.

Answer 1

𝑃(𝐴|𝐵) =
𝑃(𝐵|𝐴) 𝑃(𝐴) / 𝑃(𝐵) =
𝑃(𝐵|𝐴) 𝑃(𝐴) / 𝑃(𝐵|𝐴) 𝑃(𝐴) + 𝑃(𝐵|𝐴𝑐) 𝑃(𝐴𝑐)

Question 2

14. If events 𝐴 and 𝐵 are independent, what is the probability of observing both event 𝐴 and event 𝐵? what about event 𝐴 or event 𝐵?

Answer 2

𝑃 (𝐴 𝑎𝑛𝑑 𝐵) = 𝑃 (𝐴) × 𝑃 (𝐵)

𝑃 (𝐴 or 𝐵) = 𝑃 (𝐴) + 𝑃 (𝐵) - (𝑃 (𝐴) × 𝑃 (𝐵))

Question 3

15. How do Bayesian and Frequentist statistics differ? Explain in words and provide the relevant formulas for the probabilities that they are both calculating

Answer 3

Frequentist statistics assume a normal distribution model and use that model to calculate the probability of some data occurring:
𝑃(𝐷𝑎𝑡𝑎|𝑀𝑜𝑑𝑒𝑙).
By contrast, Bayesian Statistics involves subjective probability with a prior and aims to calculate the probability of a model occurring given some data:
𝑃(𝑀𝑜𝑑𝑒𝑙|𝐷𝑎𝑡𝑎). The latter can be precisely calculated through Bayes’ Rule:
𝑃(𝑀𝑜𝑑𝑒𝑙|𝐷𝑎𝑡𝑎) = 𝑃(𝐷𝑎𝑡𝑎|𝑀𝑜𝑑𝑒𝑙) 𝑃(𝑀𝑜𝑑𝑒𝑙) / 𝑃 (𝐷𝑎𝑡𝑎)

Question 4

18. What is a t-ratio score? Provide the formula and explain in words how it differs from a z-score. Hint: to answer this question, you will need to explain what is a z-score, provide the formula, and draw the relevant graph.

Answer 4

t = (𝑥̄ − 𝜇) / (𝑠/√𝑛)
𝑧 = 𝑋̄ − 𝜇 / 𝜎
1. Given the above formulas, the t-test substitutes the standard error, 𝑠𝑥̄, for the standard deviation, 𝜎.
2. Given the above formulas, the t-test takes into account the sample size, whereas the z-score does not.
3. The t-test involves a t-distribution, which is slightly different than a regular normal distribution, because the t-distribution depends on:
* sample size
* degrees of freedom (𝑁 − 1)
4. In practice, using the t-test will just change the statistic that we multiply in the margin of error to obtain our confidence interval.
* 𝐶𝐼 = 𝑋̄ ± 𝑀𝑂𝐸 = 𝑋̄ ± 𝑡 × 𝑠𝑋̄

Question 5

19. What are statistical significance and substantive significance?

Answer 5

Statistical significance corresponds to the p-value of usually .05 or lower, which entails rejecting the null hypothesis.
Substantive significance corresponds to the size of the coefficient value.

Question 6

20. What is the z-score associated with a 95% confidence interval?

Answer 6

1.96