Exam 9: Random Variables and Statistics

Calculate the expected value, the variance, and the standard deviation of the given random variable X. Please round your answers to two decimal places, if necessary. ​ ​X is the number of tails that come up when a coin is tossed two times. ​ $\mu = E ( X ) =$ __________ ​ $\sigma ^ { 2 } =$ __________ ​ $\sigma =$ __________

Your company issues flight insurance. You charge $2 and in the event of a plane crash, you will pay out $1 million to the victim or his or her family. In 1989, the probability of a plane crashing on a single trip was 0.00000165. If ten people per flight buy insurance from you, what was your approximate probability of losing money over the course of 110 million flights in 1989 Round your answer to four decimal places. [Hint: First determine how many crashes there must be for you to lose money.]

The following figures show the price of silver in dollars per ounce for the 10-business day period April 4 - April 15, 2004. ​ 7)0, 6.7, 7.0, 7.3, 7.0, 7.4, 7.4, 7.4, 7.3, 7.3 ​ Find the sample median. ​

Suppose X is a normal random variable with $\mu = 50$ and $\sigma = 5$ . Find the value, rounding to four decimal places, of ​ $P ( 49 \leq X \leq 58 ) =$ __________

You are performing seven independent Bernoulli trials with $p = 0.8$ and $q = 0.2$ . Calculate the probability of three successes. Please, round the answer to four decimal places.

X is a binomial variable with $n = 4$ and $p = 0.3$ . Compute $P ( 1 \leq X \leq 3 )$ . Round your answer to four decimal places if necessary.

The new computer your business bought lists a mean time between failures of 1 year, with a standard deviation of 3 months. Eight months after a repair, it breaks down again. Is this surprising (Assume that the times between failures are normally distributed.)

You are performing six independent Bernoulli trials with $p = 0.4$ and $q = 0.6$ . Calculate the probability of no successes. Round your answer to five decimal places if necessary.

X has a normal distribution with the given mean and standard deviation. Find the probability. Please, round the answer to three decimal places. ​ $\mu = 40$ , $\sigma = 5$ ​ $P ( 23 \leq X \leq 44 ) \approx$ __________

If we model after-tax household income by a normal distribution, then the figures of a 1995 study imply the information in the following table. Assume that the distribution of incomes in each country is bell-shaped and symmetric. Find the percent of canadian families which earned an after-tax income of $69,000 or more. Country U.S. Canada Switzerland Germany Sweden Mean Household Income \ 38,000 \ 35,000 \ 39,000 \ 34,000 \ 32,000 Standard Deviation \ 21,000 \ 17,000 \ 16,000 \ 14,000 \ 11,000

Suppose X is a normal random variable with $\mu = 40$ and $\sigma = 20$ . Find the value of $P ( 30 \leq X \leq 45 )$ . Please, round the answer to four decimal places.

The voltage of an electrical outlet in your home is an example of a discrete random variable.

Your company issues flight insurance. You charge $2 and in the event of a plane crash, you will pay out $1 million to the victim or his or her family. In 1989, the probability of a plane crashing on a single trip was 0.00000165. If ten people per flight buy insurance from you, what was your approximate probability of losing money over the course of 100 million flights in 1989 Round your answer to four decimal places. [Hint: First determine how many crashes there must be for you to lose money.] ?

Z is the standard normal distribution. Find the probability $P ( - 1.8 \leq Z \leq 0 )$ . Please, round the answer to four decimal places. ​ $P ( - 1.8 \leq Z \leq 0 ) \approx$ __________

Find the probability that a normal variable takes values more than $\frac { 1 } { 4 }$ standard deviations away from its mean. Please, round the answer to four decimal places.

In your bid to be elected class representative, you have your election committee survey five randomly chosen students in your class and ask them to rank you on a scale of 0 - 10. Your rankings are 5, 9, 7, 2, 3. Calculate the sample standard deviation, rounded to two decimal places.

Suppose X is a normal random variable with $\mu = 340$ and $\sigma = 20$ . Find the value of $P ( 360 \leq X \leq 370 )$ . Please, round the answer to four decimal places.

Given the data. ​ $2.4 , - 5.2,4.8 , - 0.2 , - 0.2$ ​ Compute the mean. ​ $\bar { x } =$ __________ ​ Compute the median. ​ $m =$ __________ ​ Compute the mode. ​ __________

The table shows crashworthiness ratings for several categories of motor vehicles. Take X as the crash-test rating of a small car, Y as the crash-test rating for a small SUV, and so on, as shown in the table. Overall Frontal Crash Test Rating Number Tested 3 (Good) (Acceptable) 1 (Marginal) 0(Poor) Small Cars X 16 1 11 2 2 Small SUVs Y 12 2 4 4 2 Medium SUVs 13 2 5 2 4 Passenger Vans 11 2 0 2 7 Midsize Cars V 14 2 5 0 7 Large Cars W 19 9 5 3 2 ? You choose, at random, a small car and a midsize car. What is the probability that both will be rated at most 1 Please round to three decimal places, if necessary.

Following is a sample of tow ratings (in pounds) for some popular 2000 model light trucks: 3,000, 3,000, 4,000, 5,000, 6,000, 7,000, 7,000, 7,000, 9,000, 9,000 Compute the standard deviation of the given sample. Round your answer to the nearest whole number.