question 1

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 859 hours, with a standard deviation of 94 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 (Choose one) V of the X 5 ? lifetimes lie between 624 hours and 1094 hours. (b) According to Chebyshev's theorem, at least (Choose one) V of the lifetimes lie between 671 hours and 1047 hours.There is some speculation that a simple name change can result in a shortterm increase in the price of certain business firms' stocks (relative to the prices of similar stocks). Suppose that to test the profitabilityr of name changes, we analyze the stock prices of a large sample of corporations shortly after they changed names, and we find that the mean relative increase in stock price was about 0.80%, with a standard deviation of 0.13%. Suppose that this mean and standard deviation apply to the population of all companies that changed names. Complete the following statements about the distribution of relative increases in stock price for all companies that changed names. (a) According to Chebyshev's theorem, at least of the X \\(3 9 relative increases in stock price lie between 0.54% and 1.06%. (b) According to Chebyshev's theorem, at least of the relative increases in stock price lie between 0.41% and 1.19%. A 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 48.6, with a standard deviation of 9.4. Based on this information, complete the following statements about the distribution of the scores on the writing section for the recent year. (a) According to Chebyshev's theorem, at least 56% of the scores lie X \\(3 r) between and . (Round your answer to 1 decimal place.) (b) According to Chebyshev's theorem, at least of the scores lie between 29.8 and 67.4. Fill in the P (X= x) values to give a legitimate probability distribution for the discrete random variable X, whose possible values are - 1, 0, 4, 5, and 6. Value x of X P ( X = x) - 1 0.30 0 4 0.18 6 0.15Fill in the P (X = x) values to give a legitimate probability distribution for the discrete random variable X, whose possible values are - 1, 3, 4, 5, and 6. Value x of X P ( X = x) -1 0.30 0.30 4 0.26 6Fill in the P (X=x) values to give a legitimate probability distribution for the discrete random variable X, whose possible values are 1, 3, 4, 5, and 6. Value x of X P ( X = x) 0.21 0.25 0.10 5 6 0An ordinary (fair) coin is tossed 3 times. Outcomes are thus triples of "heads" (31) and "tails" (t) which we write Path, In, etc. For each outcome, let Nbe t"ie random variable counting the number of tails in each outcome. For example, if the outcome is I\An number cube (a fair die) is rolled 3 times. For each roll, we are interested in whether the roll comes up even or odd. An outcome is represented by a string of the sort owe (meaning an odd number on the first roll, an even number on the second roll, and an even number on the third roll). For each outcome, let / be the random variable counting the number of odd rolls in each outcome. For example, if the outcome is eve, then N(eve) = 1. Suppose that the random variable X is defined in terms of N as follows: X"=2N- -4N-3. The values of X are given in the table below. Outcome ove ooo eve vee eee oeo eeo eoo Value of X -3 3 -5 -5 -3 -3 -5 -3 Calculate the probabilities P(X =x) of the probability distribution of X. 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 0 OO X P(X=x) 0 0 0 ?An number cube (a fair die) is rolled 3 times. For each roll, we are interested in whether the roll comes up even or odd. An outcome is represented by a string o the sort owe (meaning an odd number on the first roll, an even number on the second roll, and an even number on the third roll). For each outcome, let / be the random variable counting the number of even rolls in each outcome. For example, if the outcome is oee, then N (oee) = 2. Suppose that the random variable X is defined in terms of N as follows: X" = 3N-N -2. The values of X are given in the table below. Outcome 000 800 eee eeo ove deo eve vee Value of X -2 0 -2 0 0 0 0 0 Calculate the probabilities P (X =x) of the probability distribution of X. 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 5 ? P(X=x)Let X be a random variable with the following probability distribution. Value x of X P( X=x) 2 0.10 3 0.15 4 0.10 5 0.10 6 0.40 7 0.15 Complete the following. (If necessary, consult a list of formulas.)(a) Find the expectation E (X) of X. E(X (b) Find the variance Var(X) of X. Var (X