Playing Roulette In the game of roulette, a wheel consists of 38 slots numbered 0, 00, 1, 2, . . . , 36. To play the game, a ball is spun around the wheel and is allowed to fall into on of the numbered slots. If the number of the slot the ball falls into matches the number you select, you win $35; otherwise you lose $1.

1. Construct a probability distribution for the random variable X, the winnings of each spin.

2. Determine the mean and standard deviation of the random variable X.

3. Suppose that you play the game 100 times, simulate at least 5 possible outcomes. Describe the pattern.

4. Imagine a person who plays the game 1000 times a day, for 365 days. Simulate such a scenario. What is the frequency distribution of that person’s winnings and losings?

5. Suppose that you play the game 1000 times. Describe the sampling distribution of the mean amount won per game.

6. What is the probability of being ahead after playing the game 100 times?

7. What is the probability of being ahead after playing the game 200 times?

8. What is the probability of being ahead after playing the game 1000 times?

9. Based on your investigation, what lesson does this teach you? Write an essay to inform your gambler friend.

Hint: To find the mean and standard deviation:

x <- c(35, -1)
p <- c(1/38, 37/38)

(mu.X <- sum(x*p))

## [1] -0.05263158

(sigma.X <- sqrt(sum((x - mu.X)^2*p)))

## [1] 5.762617

We simulate two possible outcomes if a person plays the game 100 times.

sample(x, 100, replace = TRUE, p)

##   [1] -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1
##  [24] -1 -1 -1 -1 35 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1
##  [47] -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1
##  [70] -1 35 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 35 -1 -1
##  [93] -1 -1 -1 -1 -1 -1 -1 -1

sample(x, 100, replace = TRUE, p)

##   [1] -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1
##  [24] -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 35 -1 -1 -1 -1 -1 -1 -1 -1
##  [47] -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1
##  [70] -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1
##  [93] -1 -1 -1 -1 -1 -1 -1 -1

Repeat this simulation many times and describe the pattern.

Imagine a person who plays the game n=100n=100 times a day, for 365 days. We examine the sum:

day <- 365
daily.sum <- replicate(day,
                       sum(sample(x, 100, replace = TRUE, p)))
(tb <- table(daily.sum))

## daily.sum
## -100  -64  -28    8   44   80  116  152  188 
##   27   63   78   84   62   33   14    2    2

plot(tb)

As the number of games nn increase, the sums follow a normal distribution N(nμ,nσ2)N(nμ,nσ2). With n=1000n=1000 and class width 36,

day <- 365
daily.sum <- replicate(day,
                       sum(sample(x, 1000, replace = TRUE, p)))
(tb <- table(daily.sum))

## daily.sum
## -568 -532 -460 -424 -388 -352 -316 -280 -244 -208 -172 -136 -100  -64  -28 
##    1    1    4    4    4   12   16   16   23   17   27   26   26   24   25 
##    8   44   80  116  152  188  224  260  296  332  368  440 
##   32   19   26   16   12   11    5    4    7    3    1    3

plot(tb/day/36)
curve(dnorm(x, 1000*mu.X, sqrt(1000)*sigma.X), add = TRUE)

The distribution of sample means has a standard deviation of σ/n−−√σ/n:

daily.mean <- daily.sum/1000
hist(daily.mean, probability = TRUE,
     xlim = c(-2, 2), ylim = c(0, 2.5))
curve(dnorm(x, mu.X, sigma.X/sqrt(1000)),
      xlim = c(-2, 2), add = TRUE)

Play 100 times. What is the probability of being ahead?

n1 <- 100
curve(dnorm(x, mu.X, sigma.X/sqrt(n1)), xlim = c(-2, 2),
      ylim = c(0, 2.5), main = "n=100", ylab = "pdf")
abline(v = mu.X)
coord.x <- c(0, seq(0, 2, 0.01), 2)
coord.y <- c(0, dnorm(seq(0, 2, 0.01), mu.X, sigma.X/sqrt(n1)), 0)
polygon(coord.x, coord.y, density = 10)

1 - pnorm(0, mu.X, sigma.X/sqrt(n1))

## [1] 0.4636141

Play 200 times:

n2 <- 200
curve(dnorm(x, mu.X, sigma.X/sqrt(n2)), xlim = c(-2, 2),
      ylim = c(0, 2.5), main = "n=200", ylab = "pdf")
abline(v = mu.X)
coord.x <- c(0, seq(0, 2, 0.01), 2)
coord.y <- c(0, dnorm(seq(0, 2, 0.01), mu.X, sigma.X/sqrt(n2)), 0)
polygon(coord.x, coord.y, density = 10)

1 - pnorm(0, mu.X, sigma.X/sqrt(n2))

## [1] 0.4486139

Play 1000 times:

n3 <- 1000
curve(dnorm(x, mu.X, sigma.X/sqrt(n3)), xlim = c(-2, 2),
      ylim = c(0, 2.5), main = "n=1000", ylab = "pdf")
abline(v = mu.X)
coord.x <- c(0, seq(0, 2, 0.01), 2)
coord.y <- c(0, dnorm(seq(0, 2, 0.01), mu.X, sigma.X/sqrt(n3)), 0)
polygon(coord.x, coord.y, density = 10)

1 - pnorm(0, mu.X, sigma.X/sqrt(n3))

## [1] 0.3863597

The City of Charlotte is experiencing flooding. The City has determined that it will activate the flood

gates when the average flood level reaches 2 feet. The flood control system is activated and

resources are directed to flood control when the flood condition is equal to or more than the 2 feet

standard the City has set. The City sampled 20 spots in the urban area between 7:00am and 8:00am.

This data set will be posted to Canvas. Examine the data using the concepts you have learned in

Chapter 10. Should the City activate the flood control system? Why or why not?

Data Set:

Areas. Feet of Flood Water

1 0

2 3

3 1

4 2

5 0

6 0

7 2

8 3

9 1

10 3

11 2

12 2

13 5

14 1

15 2

16 0

17 2

18 1

19 0

20 2

Question 3

Due to the competition from Pepsi, Coca-Cola attempted to change its old recipe. Surveys of Pepsi drinkers indicated that they preferred Pepsi because it was sweeter than Coke. As a part of the analysis that led to Coke’s ill-fated move, the management of Coca-Cola performed extensive surveys in which consumers tasted various versions of the new Coke. Suppose that a random sample of 200 cola drinkers was given various versions of the Coke with different amount of sugar. After tasting the product, each drinker was asked to rate the taste quality. The possible responses were as follows:

1=poor; 2 = fair; 3 = average; 4 = good; 5 = excellent.

The responses (ratings) and sugar content (percentage by volume) of the version tested are given below

(e) Construct and interpret the 95% confidence interval for β1

(f) Predict the expected rating of Coca-Cola for the sugar level of 9 (percentage per volume).

(g) How much is the coefficient of determination and interpret it?

(h) How much is adjusted R-square? When do you use adjusted R-square?

( I ) How much is the standard error of the estimate? Is it a good model based on this criterion?

|    | a1   | a2   |
|----|------|------|
| b1 | 0.37 | 0.16 |
| b2 | 0.23 | ?    |

1. What is ?(?=?2,?=?2)P(A=a2,B=b2)?

2. Observing events from this probability distribution, what is the probability of seeing (a1, b1) then (a2, b2)?

3. Calculate the marginal probability distribution, ?(?)P(A).

4. Calculate the marginal probability distribution, ?(?)P(B).

Match the following words to choices below.

dwarfism, Addison's disease, Cushing's syndrome, hypoglycemia, type 2 diabetes, type 1 diabetes

1. increased secretions of cortisol

2. Controlled by a healthy diet and exercise

3. Insulin production decreases or fully stops

4. Increased production of adrenal corticoid hormone

5. Increased production of growth hormone

6. Elevated blood glucose levels

1. How many funds are used in state and local government accounting
2. What activities are accounted for in each of the funds identified in question #2
3. For the funds listed in question #1, how are they grouped – what are the groups/categories?
4. For the groups identified in question #3, answer the following questions:
o What is the measurement focus
o What is the accounting equation
o What is the basis of accounting used

1. What is the major byproduct of anaerobic glycolysis: ATP, NADH, lactate, or pyruvate?

2. Glucose is retained in cells because it is phosphorylated, degraded, bound, or modified by amylase?

3. Is the final product of glycolysis acetyl CoA, glucose, lactate, or pyruvate?

4. Does Glycolysis generate no ATP, 1 ATP, 2 ATPs, or NADH?

5. Are DNA ends replicated by DNA polymerase, telomerase, RNA polymerase, or primase?

1. The following table shows different baskets of black pens:

(a) List all samples of size 2, and compute the mean of each sample.

(b) Compute the mean of the distribution of the sample mean and the population mean. Compare the two values.

(c) Compare the dispersion in the population with that of the sample mean.

for each of the following scenarios:
1)discuss various techniques for communicating effectively with the client
2)give examples from personal experience if possible

scenarios:
1) you are talking with an anxious client..
2)you are talking with an angry client...
3)you are talking with a depressed client..
4)you are talking with a manipulative client..
5)you are talking with a seductive client...
6)you are talking with a client that has Aphasia...