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

A sample of size 18 will be drawn from a population with mean 4 and standard deviation 3.

(a) Is it appropriate to use the normal distribution to find probabilities for x?

(b) If appropriate find the probability that x will be greater than 3.

(c) If appropriate find the 20th percentile of x.

of nine executives in a business firm, four are married, three have never married, and two are divorced. three of the executives are to be selected for promotion. Let Y1 donate the number of married executives and Y2 donate the number of never married executives among the three selected for promotion. Assuming that three are randomly selected from nine available find the joint probability function of Y1 and Y2.

An ecologist hypothesizes that birds with shorter wing spans use wider tree branches. The ecologist captured male birds, measured their wing span and other characteristics in millimeters, and then marked and released them. During the ensuing winter, the ecologist repeatedly observed the marked birds as they foraged for food on tree branches. He noted the branch diameter on each occasion, and calculated the average branch diameter for each bird in centimeters. The measurement data are below. What can the ecologist conclude with an α of 0.01?

a) What is the appropriate statistic?
---Select--- na Correlation Slope Chi-Square
Compute the statistic selected above:

b) Compute the appropriate test statistic(s) to make a decision about H₀.
(Hint: Make sure to write down the null and alternative hypotheses to help solve the problem.)
critical value =  ; test statistic =
Decision:  ---Select--- Reject H0 Fail to reject H0

c) Compute the corresponding effect size(s) and indicate magnitude(s).
If not appropriate, input and/or select "na" below.
effect size =  ;   ---Select--- na trivial effect small effect medium effect large effect

d) Make an interpretation based on the results.

There was a significant positive relationship between the wing span of the birds and branch diameter.There was a significant negative relationship between the wing span of the birds and branch diameter.    There was no significant relationship between the wing span of the birds and branch diameter.

A passcode can have 4 or 5 digits. Digits can be repeated and leading 0s are allowed. So, 1234 would be a 4 digit code that is different from 01234, which is a 5 digit code. How many different passcodes are possible?

A population has a mean of 400 and a standard deviation of 50. Suppose a sample of size 125 is selected and x is used to estimate μ.

a. What is the probability that the sample mean will be within +/- 4 of the population mean (to 4 decimals)?

b. What is the probability that the sample mean will be within +/- 11 of the population mean (to 4 decimals)?

A random sample of 11 items is drawn from a population whose standard deviation is unknown. The sample mean is x¯ = 920 and the sample standard deviation is s = 25. Use Appendix D to find the values of Student’s t.

(a) Construct an interval estimate of μ with 95% confidence. (Round your answers to 3 decimal places.)
  
The 95% confidence interval is from to

(b) Construct an interval estimate of μ with 95% confidence, assuming that s = 50. (Round your answers to 3 decimal places.)
  
The 95% confidence interval is from to

(c) Construct an interval estimate of μ with 95% confidence, assuming that s = 100. (Round your answers to 3 decimal places.)
  
The 95% confidence interval is from to

Men’s heights are normally distributed with a mean of 69.5 inches and a standard deviation of 2.4 inches.

1. What percent of men are taller than 5 feet?

2. What percent of men can be a flight attendant if you must be between 5’2” and 6’1”?

3. Find the 45th Percentile of men’s heights.

Let X and Y have the joint PDF

f(x) = { 1/2 0 < x + y < 2, x > 0, y > 0 ;

{ 0 elsewhere

a) sketch the support of X and Y

b) Are X and Y independent? Explain.

c) Find P(x<1 and y<1.5)

Since an instant replay system for tennis was introduced a a major tournament, men challenged 1438 referee calls, with the result that 422 of the calls were overturned. Women challenged 740 referee calls, and 212 of the calls were overturned. Use a .05 significance level to test the claim that men and women have equal success in challenging calls. Complete parts (a) through (c) below. a. test the claim using a hypothesis test. Consider the first sample to be the sample of male tennis players who challenged referee calls and the second sample to be the sample of female tennis players who challenged referee calls. What are the null and alternative hypotheses for the hypothesis test? Identify the test statistic. Z= (round to 2 decimal places as needed) Identify the P-value. P= (round to 3 decimal places as needed) What is the conclusion based on the hypothesis test? The P-value is Less (than/greater) than the significance level of alpha= .05, so (fail to reject/reject) the null hypothesis. There (is sufficient/is not sufficient) evidence to warrant rejection of the claim that women and men have equal success in challenging calls. B. Test the claim by constructing an appropriate confidence interval. The 95% confidence interval is x<(p1-p2)

2. The data sample,below gives the lengths (in cm) of 43 male coyotes found in Nova Scotia. For this x=92.06ands=6.70.

78.0 80.0 90.0 90.5 96.0 96.0,80.0 81.3 83.8 84.5 85.0 86.0 86.4 86.5 87.0 88.0 88.0 88.9 88.9 91.0 91.0 91.0 91.4 92.0 92.5 93.0 93.5 95.0 95.0 95.0 95.0 95.5 96.0 96.0 97.0 98.5 100.0 100.5 101.0 101.6 103.0 104.1 105.0

This data has a bell-shaped distribution. See how well the empirical rule describes the actual distribution of this data by finding the percentage of sample values within one, two, and three standard deviations of the mean.

Q.1 A social psychologist is interested in whether the amount of couple photos (photos featuring the couple together) people post on social media would vary across relationship stages. He administered a questionnaire to a group of couples (one questionnaire for each couple) at the anniversaries of their first year, second year, third year, and fourth year of dating. The questionnaire asked about the total amount of couple photos they posted in social media during the past year. Below are the data, with a higher number denoting more couple photos posted:

a. What are the independent variable and dependent variable of this study?

b. Write down the omnibus null hypothesis and the alternative hypothesis for the overall effect of the independent variable.

c. Conduct a proper statistical test by hand calculation to test the omnibus hypothesis with 5% as the level of significance (α). (For this exercise, the data assumptions of your chosen statistical test can be taken as reasonably met.) Show your calculation formulae and steps. In case you decide to conduct an ANOVA, you are not required to conduct any post-hoc comparisons. Decide whether to reject the null hypothesis or not and state the basis of your decision.

d. Calculate the effect size in terms of eta-squared and omega-squared.

Briefly explain how Levene Test is used. Give an example to illustrate.

Among 16 electrical components exactly 4 are known not to function properly. If 6 components are randomly selected, find the following probabilities: (i) The probability that all selected components function properly. (ii) The probability that exactly 3 are defective. (iii) The probability that at least 1 component is defective.