Question

1. Multiple choice. Circle all the correct answers a) You flip a coin 100,000 times and record the outcome in a Xi 1 if the toss is "Heads" and 0 if its "Tails. The Law of Large Numbers says that: i. ii. It is impossible for the first n flips to all be "Heads" if n is large. With high probability, the share of coin flips that are "Heads" will approximate 50%. The sample mean of X is always 0.5 if the coin is fair, regardless of the number of times you flip the coin. None of the above iv. Which of the following statements are true: b. i. Type I error is the probability of rejecting the null when it is true ii. Type I error is the probability of not rejecting the null when it is true i. Type II error is the probability of rejecting the null when it is false iv. Type II error is the probability of not rejecting the null when it is false

Answer 1

a) LAW OF LARGE NUMBERS say that if an experiment is repeated independently by large number of times, then sample mean approaches theoretical mean. So, (iii) cannot be accurate.

(ii) is accurate from if we go by definition mentioned above. (i) cannot be true is n is large as the distribution becomes normal. For all first n flips to be head, the distribution is poisson distribution. But, LLN states that when n is large all distribution tend to be normal distribution.

Thus, (ii) is the only CORRECT answer.

b) (i) and (iv) are the statements that are CORRECT. For clearer understanding , we use the following example.

Consider the null hypothesis (H_o) that a person is telling truth. Thus alternative hypothesis (H_a) is the person is lying.

The court gives a verdict of either "GUILTY" or "NOT GUILTY".

	HO	HA
NOT GUILTY	RIGHT	TYPE I ERROR (FALSE POSITIVE)
GUILTY	TYPE II ERROR (FALSE NEGATIVE)	RIGHT

If the person is telling truth and court says " NOT GUILTY" then NO ERROR.

If the person is lying and court says "GUILTY" then NO ERROR.

If the person is telling truth and court says " GUILTY" then TYPE II ERROR or FALSE NEGATIVE as court failed to REJECT FALSE NULL HYPOTHESIS.

If the person is lying and court says "NOT GUILTY" then TYPE I ERROR or FALSE POSITIVE as court REJECTED TRUE NULL HYPOTHESIS.