Is formed by placing five balls in a bag The balls are labeled 1, 2, 3, 4, and 5

Question

Is formed by placing five balls in a bag The balls are labeled 1, 2, 3, 4, and 5, respectively. The mean of this population is mu = 3. Someone who does not know the contents of the bag will estimate the value of mu by randomly taking a sample of three of the balls (without replacement) and finding either the sample mean or the sample median. In a meantime, a statistician has listed all the possible samples of size three (sampling without replacement) and has calculated the sample mean and the sample median for each possible sample. The statistician finds that: All the possible sample means form a distribution whose mean is 3 and whose standard deviation is 0.577. All the possible sample medians form a distribution whose mean is 3 and whose standard deviation is 0.775. Regarding the choice between using the sample mean and using the sample median for estimating mu, which of the following is true? Both the sample mean and the sample median are unbiased, but the sample median is preferable as it has the larger standard deviation. Both the sample mean and the sample median are unbiased, but the sample mean is preferable as it has the smaller standard deviation. The sample mean is unbiased and the sample median is biased, so the sample mean is preferable. The sample median is unbiased and the sample mean is biased, so the sample median is preferable. Both the sample mean and the sample median are biased. A regular deck of cards has four suits each consisting of thirteen cards. One of the suits is "hearts" and one card in each suit is a "queen". One card is going to be picked at random from the deck. Let A be the event that the card is a heart and let B be the event that the card is a queen. Which of the following is true? The events A and B are independent and mutually exclusive. The events A and B are independent but not mutually exclusive. The events A and 5 are not independent but are mutually exclusive. The events A and B are not independent and not mutually exclusive. It is not possible to tell from the information given whether or not live events A and B are mutually exclusive.

Explanation / Answer

33:

Out of 52 cards 13 cards are heart so the probability of getting a heart is

P(A) = 13 /52

Out of 52 cards, 4 are queen so the probability of getting a queen is

P(B) = 4/52

Out of 52 cards, one card is queen of heart so events A and B are not mutually exclusive.

Now

P(A and B) = 1/52

Since

P(A and B) = P(A)P(B)

So event A and B are independent.

Hence, option B is true.

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