Exam 9: Random Variables and Statistics

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 ( Good) (Acceptable) (Marginal) (Poor) Small Cars 16 1 11 2 2 Small SUVs 12 2 4 4 2 Medium SUVs 15 3 5 3 4 Passenger Vans 13 3 0 3 7 Midsize Cars V 15 3 5 0 7 Large Cars W 19 9 5 3 2 ​ Find the category that has the lowest probability of a Good rating.

Compute the (sample) standard deviation of the data sample. Round your answer to the nearest whole number, if necessary. ​ $- 2,5,7,8,13$ ​ $s =$ __________

This exercise is based on the following information, gathered from student testing of a statistical software package called MODSTAT. Students were asked to complete certain tasks using the software, without any instructions. The results were as follows. (Assume that the time for each task is normally distributed.) It can be shown that if X and Y are independent normal random variables with means $\mu _ { X }$ and $\mu _ { y }$ and standard deviations $\sigma _ { X }$ and $\sigma _ { y }$ respectively, then their sum $X + Y$ is also normally distributed and has mean $\mu = \mu _ { X } + \mu _ { Y }$ and standard deviation $\sigma = \sqrt { \sigma ^ { 2 } X + \sigma ^ { 2 } Y }$ . Assuming that the time it takes a student to complete each task is independent of the others, find the probability that a student will take at least 20 minutes to complete both Tasks 3 and 4. Round your answer to four decimal places. Task Mean Time (minutes) Standard Deviation Task 1: Descriptive Analysis of Data 11.4 5.0 Task 2: Standardizing Scores 11.9 9.0 Task 3: Poisson Probability Table 7.5 4.1 Task 4: Areas Under Normal Curve 9.5 5.9

You are performing 5 independent Bernoulli trials with $p = 0.6$ and $q = 0.4$ . Calculate the probability that it will be at most 2 successes. Round your answer to five decimal places if necessary.

Kent's Tents has 5 green knapsacks and 3 yellow ones in stock. Curt selects 3 of them at random. Let X be the number of green knapsacks he selects. Find $P ( X \leq 2 )$ .

Let X be the lower number when two dice are rolled. Calculate the expected value of the given random variable X. Please, round your answer to the nearest thousandth.

Select five cards without replacement from a standard deck of 52, and let X be the number of red cards you draw. Calculate the expected value of the given random variable X. ​ $E ( X ) =$ __________

The population standard deviation is greater than the sample standard deviation.

Find the expected value of a random variable X having the following probability distribution: x 2 4 6 8 P(X=x) Round your answer to tenth if necessary.

LSAT test scores are normally distributed with a mean of 500 and a standard deviation of 100. Find the probability that a randomly chosen test taker will score 340 or lower. Round your answer to four decimal places.

Your class is given a mathematics exam worth 150 points; X is the average score, rounded to the nearest whole number. ​ Classify the random variable X as finite, discrete infinite, or continuous. ​

In some year, 21 percent of all teenagers in the U.S. had checking accounts. Your bank, TeenChex Inc., is interested in targeting teenagers who do not already have a checking account. ​ TeenChex selects a random sample of 1,000 teenagers. Find the interval in which the chance that teenagers in the sample will not have checking accounts is approximately 95 percent. Please round your answer to the nearest whole number. ​

Give the probability distribution for the indicated random variable. A fair coin is tossed three times, and we count the number of "runs" that occur. A "run" is defined as a sequence of one or more of the same letter (e.g., HTH is 3 runs, TTH is 2 runs, and TTT is 1 run).

According to quantum mechanics, the energy of an electron in a hydrogen atom can assume only the values $\frac { k } { 1 }$ , $\frac { k } { 4 }$ , $\frac { k } { 9 }$ , $\frac { k } { 16 }$ , . . . for a certain constant value $k$ . X - the energy of an electron in a hydrogen atom. Classify the random variable X as finite, discrete infinite, or continuous.

Give the probability distribution for the indicated random variable. A black and white die are rolled, and consider the following random variable. $X = \left\{ \begin{array} { l } 0 , \text { if both numbers are odd } \\3 , \text { if one number is odd and one is even } \\6 , \text { if both numbers are even }\end{array} \right.$

X is a binomial variable with $n = 6$ and $p = 0.4$ . Compute $P ( X = 5 )$ . Round your answer to four decimal places. ​ $P ( X = 5 ) =$ __________

A roulette wheel has the numbers 1 through 36, 0, and 00. A bet on two numbers pays 17 to 1 (that is, if one of the two numbers you bet comes up, you get back your $1 plus another $17). How much do you expect to win with a $10 bet on two numbers Please, round your answer to the nearest thousandth of a dollar. ?

This exercise is based on the following information, gathered from student testing of a statistical software package called MODSTAT. Students were asked to complete certain tasks using the software, without any instructions. The results were as follows. (Assume that the time for each task is normally distributed.) It can be shown that if X and Y are independent normal random variables with means $\mu _ { X }$ and $\mu _ { y }$ and standard deviations $\sigma _ { X }$ and $\sigma _ { y }$ respectively, then their sum $X + Y$ is also normally distributed and has mean $\mu = \mu _ { X } + \mu _ { Y }$ and standard deviation $\sigma = \sqrt { \sigma ^ { 2 } X + \sigma ^ { 2 } Y }$ . Assuming that the time it takes a student to complete each task is independent of the others, find the probability that a student will take at least 20 minutes to complete both Tasks 3 and 4. Round your answer to four decimal places. Round Z to two decimal places. Task Mean Time ( minutes ) Standard Deviation Task 1: Descriptive Analysis of Data 11.4 5.0 Task 2: Standardizing Scores 11.9 9.0 Task 3: Poisson Probability Table 7.3 3.9 Task 4: Areas Under Normal Curve 9.1 5.5

The probability that a randomly chosen citizen-entity of Cygnus is of pension age is approximately 0.7. What is the probability that, in a randomly selected sample of 3 citizen-entities, all of them are of pension age Round your answer to four decimal places.

Find the expected value of a random variable X having the following probability distribution: x 10 20 30 40 P(X=x) Round your answer to tenth if necessary.