Q1.

BIG Corporation produces just about everything but is currently interested in the lifetimes of its batteries. To investigate its new line of Ultra batteries, BIG randomly selects 1000 Ultra batteries and finds that they have a mean lifetime of 863 hours, with a standard deviation of 90 hours. Suppose that this mean and standard deviation apply to the population of all Ultra batteries. Complete the following statements about the distribution of lifetimes of all Ultra batteries.

(a)According to Chebyshev's theorem, at least ?

36%/56%/56%/75%/84%/89%

of the lifetimes lie between 683 hours and 1043 hours.

(b)According to Chebyshev's theorem, at least

56% of the lifetimes lie between hours and hours.

(Round your answer to the nearest whole number.)

Q2.

Fill in the P(X=x) values to give a legitimate probability distribution for the discrete random variableX, whose possible values are 0, l, 4, 5, and 6. Value x of X Fill in the P (X= x) values to give a legitimate probability distribution for the discrete random variable ), whose possible values are 1, 2, 4, 5, and 6. Value x of X P( X = x) 0.22 2 4 0.19 5 0.10 6Fill in the P (X = x) values to give a legitimate probability distribution for the discrete random variable X, whose possible values are 2, 3, 4, 5, and 6. Value x of X P( X =x) 2 0.12 W 0.22 4 U 0.10 6 OSuppose that the genders of the three children of a certain family are soon to be revealed. Outcomes are thus triples of "girls" (g) and "boys" (b), which we write gbg, bbb, etc. For each outcome, let R be the random variable counting the number of girls in each outcome. For example, if the outcome is bgg, then R(bgg) = 2. Suppose that the random variable X is defined in terms of R as follows: X=6R-2R- -2. The values of X are given in the table below. Outcome bbg ggb bgg bgb gbb bbb gog ggg Value of X 2 2 2 2 2 -2 2 -2 Calculate the values of the probability distribution function of X, i.e. the function p . First, fill in the first row with the values of X. Then fill in the appropriate probabilities in the second row. Value X of X X ? Px (x)Suppose that the genders of the three children of a certain family are soon to be revealed. Outcomes are thus triples of "girls" (g) and "boys" (b), which we write gbg, bbb, etc. For each outcome, let R be the random variable counting the number of girls in each outcome. For example, if the outcome is gob, then R (gbb) = 1. Suppose that the random variable X is defined in terms of R as follows: X = R- -2R -4. The values of X are given in the table below. Outcome gbg bbg bbb ggb gbb bgb 888 bgg Value of X -4 -5 -4 -4 -5 -5 -4 Calculate the values of the probability distribution function of Y, i.e. the function p . First, fill in the first row with the values of Y. Then fill in the appropriate probabilities in the second row. Value X of X X ? Px (x) 0An ordinary (fair) coin is tossed 3 times. Outcomes are thus triples of "heads" (h) and "tails" (1) which we write hth, ttt, etc. For each outcome, let R be the random variable counting the number of heads in each outcome. For example, if the outcome is hhh, then R (hhh) = 3. Suppose that the random variable X is defined in terms of R as follows: X =2R- -6R-1. The values of X are given in the table below. Outcome ttt tth hth thh hht tht htt hhh Value of X -5 -5 -5 -5 -5 -5 - 1 Calculate the values of the probability distribution function of X, i.e. the function p . First, fill in the first row with the values of Y. Then fill in the appropriate probabilities in the second row. Value X of X 0 X Px (x) ?Let X be a random variable with the following probability distribution. Value x of X P(X=x) -20 0.25 -10 0.05 0 0.55 10 0.05 20 0.05 30 0.05 Complete the following. (If necessary, consult a list of formulas.) (a) Find the expectation E (X) of X. E(X) = 0 x ? (b) Find the variance Var(X) of X. Var(X) = 0A nationwide test taken by high school sophomores and juniors has three sections, each scored on a scale of 20 to 80. In a recent year, the national mean score for the writing section was 51.9, with a standard deviation of 9.9. Based on this information, complete the following statements about the distribution of the scores on the writing section for the recent year. 8 o (a) According to Chebyshev's theorem, at least 3 (about 89%) of the X \\(3 . scores lie between I] and I]. (Round your answer to 1 decimal place.) (Choose one) V of the (b) According to Chebyshev's theorem, at least scores lie between 32.1 and 71.7