Problem 2.

You pay 5$ /round (nonrefundable) to play the game of rolling a pair of fair dice.

If you roll an even sum, you lose, no pay off. If you roll an odd sum, that's your win (say, roll of 7 pays you 7$).

Discrete random variable X represents the winnings.

• For example, the lowest value of X is x=0 when you roll an even sum.
• For example, x=3 only when you roll {1,2} or {2,1}. You win 1+2=3$ (odd sum). AndP ( x = 3 ) = P ( { 1 , 2 } ) + P ( { 2 , 1 } ) = 1 / 36 + 1 / 36 = 1 / 18.

a) Find all possible values of X with their probabilities. Make the table as in Problem 1, a) for the probability distribution of X. Above, we calculated just one row of the table:

Dice related probabilities are discussed in 5.1, page 252, Example 5.

b) Find the expected value of X and interpret it.

Expected value is discussed in 6.1, page 320 (as mean), 321 and Examples 5,6,7.

c) Does it make mathematical sense to play the game? Remember, you have to pay 5$/game to play, what is your net gain/loss per game in the long run?

d) What price a (instead of 5$) would make the game fair? It is called the fair price as you break even in the long run: μ ( X ) − a = 0.

I do not know why I keep getting this question wrong, I triple checked my work!

A distribution of values is normal with a mean of 80 and a standard deviation of 18. From this distribution, you are drawing samples of size 23.

Find the interval containing the middle-most 40% of sample means: ANSWER HERE

Enter your answer using interval notation. In this context, either inclusive or exclusive intervals would be acceptable. Your numbers should be accurate to 1 decimal places. Answers obtained using exact z-scores or z-scores rounded to 3 decimal places are accepted.

I got 84.5 as my answer

An article reported that what airline passengers like to do most on long flights is rest or sleep; in a survey of 3697 passengers, almost 80% did so. Suppose that for a particular route the actual percentage is exactly 80%, and consider randomly selecting nine passengers. Then x, the number among the selected nine who rested or slept, is a binomial random variable with n = 9 and p = 0.8. (Round your answers to four decimal places.)

(a) Calculate p(6).

b) Calculate p(9), the probability that all nine selected passengers rested or slept.
p(9) =  

(c) Determine P(x ≥ 6).

Exercise 4 (Indicator variables). Let (Ω, P) be a probability space and let E ⊆ Ω be an event. The indicator variable of the event E, which is denoted by 1E , is the RV such that 1E (ω) = 1 if ω ∈ E and 1E(ω)=0ifω∈Ec.Showthat1E isaBernoullivariablewithsuccessprobabilityp=P(E).

Exercise 5 (Variance as minimum MSE). Let X be a RV. Let xˆ ∈ R be a number, which we consider as a ‘guess’ (or ‘estimator’ in Statistics) of X . Let E[(X − xˆ)2] be the mean squared error (MSE) of this estimation.

(i) Showthat

E[(X −xˆ)2]=E[X2]−2xˆE[X]+xˆ2 (2) =(xˆ−E[X])2 +E[X2]−E[X]2 (3) =(xˆ−E[X])2 +Var(X). (4)

(ii) ConcludethattheMSEisminimizedwhenxˆ=E[X]andtheglobalminimumisVar(X).Inthis sense, E[X ] is the ‘best guess’ for X and Var(X ) is the corresponding MSE.

Exercise 6. Suppose we have the following sample of Google’s stock price for the past 50 weeks (unit in $ per stock).

320 326 325 318 322 320 329 317 316 331 320 320 317 329 316 308 321 319 322 335 318 313 327 314 329 323 327 323 324 314 308 305 328 330 322 310 324 314 312 318 313 320 324 311 317 325 328 319 310 324

(i) Compute the sample mean x ̄ and sample standard deviation x.
(ii) Draw the ordered stem-and-leaf display. How many sample values are between x ̄ ± s, and

x ̄±2s?
(iii) Give the five-number summary of the sample. Draw the corresponding box plot.

First-born children tend to develop language skills faster than their younger siblings. One possible explanation is that they have undivided attention from their parents. If this is correct, then triplets should be slower. The following hypothetical data demonstrate potential measure of language skill from such a study. Do the data indicate significant differences? Show all work.

Single Child: 8, 7, 10, 6, 9

Triplet: 4, 4, 7, 2, 3

Determine the margin of error for a 99​% confidence interval to estimate the population mean when s​ = 43 for the sample sizes below. ​a) n=12 ​b) n=25 ​c) n=46 ​a) The margin of error for a 99​% confidence interval when n=12 is _.

A product is being assembled and packaged on three production lines (line A, line B, and line C). Each day, the quality control team selects a production line and inspects a batch chosen at random from the output of the selected production line. Line A is selected with probability .5, line B is selected with probability .2, and line C is selected with probability .3. The probability that no defects will be found in a batch selected from line A is 0.98, and the corresponding probabilities for production lines B and C are 0.96 and 0.94, respectively.

a) What is the probability that the quality control team inspects a batch from line A and finds no defects?

b) What is the probability that the quality control team inspects a batch from line B and finds no defects?

c) What is the probability that the quality control team inspects a batch from line C and finds no defects?

d) What is the probability that no defects are found in any given day?

e) Given that no defects were found in a given day, what is the probability the inspected batch came from line A?

f) Given that no defects were found in a given day, what is the probability the inspected batch came from line B?

g) Given that no defects were found in a given day, what is the probability the inspected batch came from line C?

what is the answer for question 4 Determine who placed 20^th in the men’s and the women’s 10,000 meter races at the 2016 Summer Olympics in Rio de Janeiro. Record the names, nationalities, and times. Be sure to properly cite at least two sources on a Works Cited page. Under “assignment specifications” below, there are some guidelines to help you with citing your sources.

The excel data are for the mean body and brain temperature of six ostriches was recorded at typical hot conditions. The results, in degrees Celsius.

Ostriches live in hot environments, and they are normally exposed to the sun for long periods. Mammals in similar environments have special mechanism for reducing the temperature.

1. Test for a mean difference in temperature between body and brain for these ostriches. Paste the statistics output from excel
2. Find the test statistic, p-value
3. Interpret the results to a person who do not know statistics

Data:

match vocab with definition

Box A contains 6 red balls and 3 green balls, whereas box B contains 3 red ball and 15 green balls.

Stage one. One box is selected at random in such a way that box A is selected with probability 1/5 and box B is selected with probability 4/5.

Stage two. Finally, suppose that two balls are selected at random with replacement from the box selected at stage one.

g) What is the probability that both balls are red?

h) Given that both balls are red, what is the probability they came from box A?

i) What is the probability that one ball is red and the other is green?

j) Given that one ball is red and the other is green, what is the probability they came from box A?

Terri Vogel, an amateur motorcycle racer, averages 129.99 seconds per 2.5 mile lap (in a 7 lap race) with a standard deviation of 2.25 seconds . The distribution of her race times is normally distributed. We are interested in one of her randomly selected laps. (Source: log book of Terri Vogel) Let X be the number of seconds for a randomly selected lap. Round all answers to 4 decimal places where possible. a. What is the distribution of X? X ~ N( , ) b. Find the proportion of her laps that are completed between 127.18 and 129.89 seconds. c. The fastest 2% of laps are under seconds. d. The middle 40% of her laps are from seconds to seconds.

3. Box A contains 6 red balls and 3 green balls, whereas box B contains 3 red ball and 15 green balls.

Stage one:One box is selected at random in such a way that box A is selected with probability 1/5 and box B is selected with probability 4/5.

Stage two: First, suppose that 1 ball is selected at random from the box selected at stage one.

a) What is the probability that the ball is red?

b) Given that the ball is red, what is the probability it came from box A? Next, suppose that two balls are selected at random without replacement from the box selected at stage one.

c) What is the probability that both balls are red?

d) Given that both balls are red, what is the probability they came from box A?

e) What is the probability that one ball is red and the other is green?

f) Given that one ball is red and the other is green, what is the probability they came from box A?

1. Imelda has a large collection of shoes. She has 20 pairs of sneakers of which: 9 pairs are canvas and 11 are leather; 8 pairs are white, 7 are black, and 5 are red; 5 pairs are white and canvas, 1 pair is black and canvas, and 2 pairs are red and leather. Suppose Imelda selects a pair of sneakers at random from this collection of 20 pairs.

a) What is the probability that she selects a red canvas pair?

b) Given that the pair she selected is canvas, what is the probability that they are red?

c) Given that the pair she selected is red, what is the probability that they are canvas?

d) What is the probability that she selects a white leather pair?

e) Given that the pair she selected is white, what is the probability that they are leather?

f) Given that the pair she selected is leather, what is the probability that they are white?