An Olympic archer misses the​ bull's-eye 13​% of the time. Assume each shot is independent of the others. If she shoots 9 ​arrows, what is the probability of each of the results described in parts a through f​ below? ​a) Her first miss comes on the fourth arrow. The probability is 9.8. ​(Round to four decimal places as​ needed.) ​b) She misses the​ bull's-eye at least once. The probability is 0.6731. ​(Round to four decimal places as​ needed.) ​c) Her first miss comes on the second or third arrow. The probability is nothing. ​(Round to four decimal places as​ needed.) ​d) She misses the​ bull's-eye exactly 3 times. The probability is nothing. ​(Round to four decimal places as​ needed.) ​e) She misses the​ bull's-eye at least 3 times. The probability is nothing. ​(Round to four decimal places as​ needed.) ​f) She misses the​ bull's-eye at most 3 times. The probability is nothing. ​(Round to four decimal places as​ needed.)

A few years ago, a number of prominent news sources published articles with provocative titles such as “You Now Have a Shorter Attention Span Than a Goldfish” (Time Magazine, 5/14/2015). These articles referenced a Microsoft study that claimed: • The average attention span of humans was 12 seconds in 2000. • The average attention span of humans was 8 seconds in 2013. • The average attention span of a goldfish is 9 seconds. Suppose all attention spans are normally distributed with a standard deviation of 2.1 seconds. (a) Assuming the numbers above are accurate, what is the probability a random person’s attention span in 2013 was greater than 9.5 seconds? (b) What is the probability that a randomly selected group of ten people in 2013 had an average attention span greater than 9.5 seconds? (c) A random group of 16 Denison students is found to have an average attention span of 10.3 seconds. Perform a hypothesis test to determine if this is statistically significant evidence at the α = 0.05 level to conclude Denison students have a longer attention span than a goldfish. (d) Are there any problems with the reported study? Do some internet searching to find problems with the figures presented.

General Electric (GE) appliances come with a traditional warranty, but the company also offers extended coverage with a Service Protection Advantage plan from Assurant, GE's authorized provider of extended warranties. The plan covers parts and labor for appliance repairs for an extended period. According to data compiled by Assurant, thirty percent of consumers who purchase a new GE appliance also buy the Service Protection Advantage plan.

Assume that a random sample of fourteen consumers who purchased a new GE appliance is drawn. Let us define X to be a binomial random variable representing whether or not a consumer purchased the Service Protection Advantage plan with their new GE appliance purchase this year.

a) What is the probability that no more than six of the sampled consumers purchased the Service Protection Advantage plan with their new GE appliance?

b) What is the probability that at least seven, but less than ten, of the sampled consumers purchased the Service Protection Advantage plan with their new GE appliance?

c) If you were to take repeated random samples of fourteen consumers who purchased a new GE appliance, what is the average number of them you would expect to also purchase the Service Protection Advantage plan?

Think of some discrete random variable you observe on a regular basis. For example, it could be the (rounded) number of hours you sleep, how many gallons of gas are in your car when you get into it, how many boxes of cereal are in your house, how many days between grocery shopping, etc. (just make sure it takes only integer values). Try to list all of the possible values that this discrete random variable can take. If you can, collect some frequency data – give the relative frequency table and use this as an estimate of the probability distribution. Calculate the expected value and the standard deviation for this probability distribution. Interpret these parameters, and discuss whether they make sense based on your experience. Week 4 Responses (100+ words, x2): Look at your classmates’ distribution. Is there any well-known distribution that could be used to model their random phenomenon? Some well-known discrete distributions are: Uniform, Bernoulli, Binomial, Geometric, and Poisson (but there are othbe appropriate. Post a picture of tdiscrete distribution and a histogram of the frequency data from the original post, and comment on what is similar and different. Are there any outliers that, if removed, would make the frequency data match the distribution really well?

1. Create the probability distribution in a table for all the outcomes where X is the random variable representing the number of points awarded. (already done)

2. Communicate how you arrived at the probability of each outcome.

3. What is the expected value, E(X), for the game? You may include this in your table from the distribution. If the game costs 10 points to play, how much would the player expect to win or lose?

Three Prize roller

Word scramble

a)

Ten spinner

Here, the basic (Harris-Todaro) model, also shown in the class slides, is shown:  

M = β(W_U^e – W_R), where

M = migration from the rural to the urban sector

W_U^e= the expected urban wage = pW_U, where W_U is the actual urban wage,

p is the probability of finding an urban job, i.e., the employment rate measured as E/(E+U),

where E = employed and U = unemployed.

W* = the subsistence wage in the rural sector that exists in the present of surplus labor.  

W_R = the actual rural wage, which = w* in the presents of surplus labor.

β = a coefficient that represents the responsiveness of migration to rural-urban wage disparities, e.g. if the hukou (household registration) system is

      rigorously enforced, then β is small).

Now use the migration model to explain the following (a diagram is optional):

1. The known agricultural wage is 10. With robust productivity growth, the industrial wage rises to 20.   Rural workers anticipate that if they migrate, the fixed probability of finding a job is 75%.  How is the new equilibrium achieved?

2. Now suppose that the economy is initially in equilibrium. The Chinese government raises the minimum wage in the urban-industrial sector by 20%. However, the number of jobs in the urban industrial sector is fixed.  How is the new equilibrium achieved?

3. The number of coins that Jake spots when walking to work can be modeled as a Poisson process with unit rate of 3/half an hour. Suppose it takes Jake half an hour each day to walk to work, and half an hour to walk back. Each coin is equally likely to be a penny, a nickel, a dime, or a quarter. Jake ignores the pennies but picks up the other coins.

   (a) Find the expected amount of money that Jake picks up on his way to work. (b) Find the variance of the amount of money that Jake picks up on a given day. (c) Find the probability that Josh picks up exactly 5 cents on his way to work. You do not have to calculate this. Just provide an expression.

Suppose you play 1000 spins of Roulette and on each time you bet that the ball will land on a red number

3. Use the normal approximation to compute the probability of winning

a. More than 470 times: P(X > 470)? (5-pts)

b. P (X<475)? (5-pts)

c. P(X>500)? (5-pts) i. Note winning more than 500 times is necessary to walk away a ‘winner’ if you bet the same amount each time.

d. P(X<500)? (5pts) i. Note winning less than 500 times means you walk away a ‘loser’ if you bet the same amount each time.