In this project we will consider a particle sitting at the origin inside of a square whose sides intersect at the points (-10, -10), (-10, 10), (10, -10), and (10, 10). Each second, the particle moves in a random direction a random distance. That distance has an X as well as a Y component, each of which are standard normally distributed.

1.1 [8 points] Use R-Studio to model the following scenario: the particle moves every second as described above, use your code to model that motion until the particle steps out of bound “FOR THE FIRST TIME” then stop counting the steps.

1.2 [5 points] Repeat this simulation 1,000 times and store the number of steps in a matrix, then calculate the mean number of steps needed for the particle to step out of bounds, the median and the standard deviation.

1.3 [4 points] Plot a box plot and a histogram of the number of steps of all replications.

1.4 [3 points] Using the last simulation run, trace the motion of that particle using a line plot. What where the coordinates of the particle when it stepped out of bound.

1. a) You’re a marketing analyst for Hasbro Toys. You get the following data:

   Compute and interpret the sample correlation coefficient between advertising expenditure and sales revenue.

b) An experiment results in one of the following sample points: E₁, E₂, E₃, E₄, or E₅.

    Find P(E₃) if P(E₁) = 2P(E₃), P(E₂₎= 0.1, P(E₄) =0 .2 and P(E₅) = 0.1

c) For each of the random variables defined below, what values may each of the random variables X assume?

(i) (1 point) X=the number of newspapers sold by the New York Times each month

(ii) (1 point) X= amount of ink used in printing the Sunday edition of the New York Times.

A company that packages salted peanuts in 12-ounce jars is interested in checking how well one of its machines is functioning. Proper functioning would mean averaging 12 ounces of peanuts per jar and not consistently over or under-filling the jars. To test the machine, a sample of 100 jars is taken from the line at random time intervals and their contents weighted. The sample mean weight of a jar turns out to be 12.5 ounces. Using a significance level of α = 0:05, please check if an adjustment is necessary for the machine filling the jars. Explicitly state your hypothesis, go through all steps and make sure you make a recommendation at the end. Conduct the hypothesis test assuming that σ 2 is known to be equal to 6:4. Perform the hypothesis test by:

a) looking at the observed value vs the critical value.

b) computing the p-value and comparing it to α = 0:05.

c) constructing a 95% confidence interval. Make sure you write the hypotheses tests and the test statistic.

A sample space has four possible discrete outcomes: S={1,2,3,4} with probabilities 0.1, 0.2, 0.3, 0.4 respectively.
a) Sketch the density function fx(x)
b) Write the equation for the density function
c) Calculate the probability of outcomes between 2 and 3 inclusively
d) Sketch the distribution function Fx(x)
e) Write the equation of the distribution function
f) Use the distribution function to calculate the probability of outcomes between 2 and 3 inclusively (don't forget to use the next lower outcome for the lower limit)

In 1995, a random sample of 100 adults that had investments in the stock market found that only 20 said they were investing for the long haul rather than to make quick profits. A simple random sample of 100 adults that had investments in the stock market in 2002 found that 36 were investing for the long haul rather than to make quick profits. Let p1995 and p2002 be the actual proportion of all adults with investments in the stock market in 1995 and in 2002, respectively, that were investing for the long haul.

a. What are the 1995 and 2002 estimated proportions of all adults with long haul investments in the stock market?

b. State the null and alternative hypotheses if the researcher is interested to see if nothing changed when it comes to the proportions of adults investing for the long haul between 1995 and 2002. 6

c. Conduct the hypothesis test from point b. at α=0.05 and formulate the conclusion using the observed vs critical value method.

d. State the null and alternative hypotheses for the case in which the researcher wants to test if the proportion of adults investing for the long haul increase over time. e. Conduct the hypothesis test from point d. at α=0.05 using the p-value method.

f. Construct the 95% confidence interval for the difference in proportions associated with the hypothesis test from b.

g. What is the margin of error for a 97% confidence interval for the difference in proportions associated with the hypothesis test from b? How does.it compare with the margin of error from point f. Explain the differences.

Question (5) [12 Marks] Note: Do not use R, do the calculations by hand.

A very large (essentially infinite) number of butterflies is released in a large field. Assume the butterflies are scattered randomly, individually, and independently at a constant rate with an average of 6 butterflies on a tree.

(a) [3 points] Find the probability a tree (X) has > 3 butterflies on it.  

(b) [3 points] When 10 trees are picked at random, what is the probability 8 of these trees have > 3 butterflies on them?

(c) [3 points] Find the probability a tree with > 3 butterflies on it has exactly 6.

(d) [3 points] On 2 trees there are a total of t butterflies. Find the probability that x of these butterflies are on the first tree. Note: Use the conditional probability to solve this part.

Assume that you are hired by the Governor’s Office to study whether a tax on liquor has affected alcohol consumption in the state. You are able to obtain, for a sample of fifty individuals selected at random, the difference in liquor consumption (in ounces) for the year before and after the tax. For person i who is sampled randomly from the population, Yi denotes the change in liquor consumption. Treat these as a random sample from a Normal(µ, σ2 ) distribution.

(A) The Governor believes there was no change in alcohol consumption. How do you test his claim? State the null and alternative hypotheses.

(B) What if the Governor thought there was actually a drop in liquor consumption as a result of the tax? How would you setup the test in this case? State the null and alternative hypotheses. What economic arguments would you use to justify your choice of hypothesis test? 4

(C) Now assume that for your sample of size n=50, you have obtained a mean ?̅ = −32.8 and a sample standard deviation s = 100. Formally conduct the hypothesis test from part (B) using these values and a 5% significance level. Does your conclusion change at a 1% significance level?

(D) Construct a 95% confidence interval for the mean change in liquor consumption based on your hypothesis from point (A).

In a certain region, 10 percent of the homes have solar panels. A city official is investigating energy consumption for homes within the region. Each week, the city official selects a random sample of homes from the region. Let random variable Y represent the number of homes selected at random from the region until a home that has solar panels is selected. The random variable Y has a geometric distribution with a mean of 10. Which of the following is the best interpretation of the mean?

A. Each week, the number of homes with solar panels increases by 10.

B. For a randomly selected week, it will take 10 homes before a home with solar panels is selected.

C. The average number of solar panels per home is equal to 10.

D. Over many weeks, the average number of homes with solar panels is 10.

E. Over many weeks, it takes 10 homes, on average, before a home with solar panels is selected.

Dorothy Kelly sells life insurance for the Prudence Insurance Company. She sells insurance by making visits to her clients homes. Dorothy believes that the number of sales should depend, to some degree, on the number of visits made. For the past several years, she kept careful records of the number of visits (x) she made each week and the number of people (y) who bought insurance that week. For a random sample of 15 such weeks, the x and y values follow.

x 11 20 15 12 28 5 20 14 22 7 15 29 8 25 16

y 3 12 10 5 8 2 5 6 8 3 5 10 6 10 7

In this setting we have Σx = 247, Σy = 100, Σx2 = 4839, Σy2 = 790, and Σxy = 1873.

(a) Find x, y, b, and the equation of the least-squares line. (Round your answers for x and y to two decimal places. Round your least-squares estimates to four decimal places.)

x =

y =

b =

ŷ = + x

(b) Draw a scatter diagram displaying the data. Graph the least-squares line on your scatter diagram. Be sure to plot the point (x, y).

(c) Find the sample correlation coefficient r and the coefficient of determination. (Round your answers to three decimal places.)

r =

r2 =

What percentage of variation in y is explained by the least-squares model? (Round your answer to one decimal place.) %

(d) Test the claim that the population correlation coefficient ρ is positive at the 1% level of significance. (Round your test statistic to three decimal places.)

t =

Find or estimate the P-value of the test statistic.

P-value > 0.250

0.125 < P-value < 0.250

0.100 < P-value < 0.125

0.075 < P-value < 0.100

0.050 < P-value < 0.075

0.025 < P-value < 0.050

0.010 < P-value < 0.025

0.005 < P-value < 0.010

0.0005 < P-value < 0.005

P-value < 0.0005

Conclusion

Reject the null hypothesis. There is sufficient evidence that ρ > 0.

Reject the null hypothesis. There is insufficient evidence that ρ > 0.

Fail to reject the null hypothesis. There is sufficient evidence that ρ > 0.

Fail to reject the null hypothesis. There is insufficient evidence that ρ > 0.

(e) In a week during which Dorothy makes 20 visits, how many people do you predict will buy insurance from her? (Round your answer to one decimal place.)

_____ people

(f) Find Se. (Round your answer to three decimal places.)

Se =

(g) Find a 95% confidence interval for the number of sales Dorothy would make in a week during which she made 20 visits. (Round your answers to one decimal place.)

lower limit sales

upper limit sales

(h) Test the claim that the slope β of the population least-squares line is positive at the 1% level of significance. (Round your test statistic to three decimal places.)

t =

Find or estimate the P-value of the test statistic.

P-value > 0.250

0.125 < P-value < 0.250

0.100 < P-value < 0.125

0.075 < P-value < 0.100

0.050 < P-value < 0.075

0.025 < P-value < 0.050

0.010 < P-value < 0.025

0.005 < P-value < 0.010

0.0005 < P-value < 0.005

P-value < 0.0005

Conclusion

Reject the null hypothesis. There is sufficient evidence that β > 0.

Reject the null hypothesis. There is insufficient evidence that β > 0.

Fail to reject the null hypothesis. There is sufficient evidence that β > 0.

Fail to reject the null hypothesis. There is insufficient evidence that β > 0.

(i) Find an 80% confidence interval for β and interpret its meaning. (Round your answers to three decimal places.)

lower limit

upper limit

Interpretation

For each additional visit made, sales increase by an amount that falls within the confidence interval.

For each less visit made, sales increase by an amount that falls within the confidence interval.

For each additional visit made, sales increase by an amount that falls outside the confidence interval.

For each less visit made, sales increase by an amount that falls outside the confidence interval.

Wage and unions. Listed below are sample characteristics from a 1987 survey that examines average hourly wage rates for union and non-union workers. Nonunion: ?̅?? = 11.47; ???= 1206; ???= 6.58 Union: ?̅? = 12.19; ??= 376; ?? = 4.77

a. What is the difference in average hourly wages between union and nonunion workers?

b. Construct a 95% confidence interval around this difference.

c. Test the null hypothesis that there is no difference in wages across the two groups.

Please interpret the coefficients of the following estimated regressions:

(a) rdintens ̂ = 2:625 + 0:000053sales. Note that rdintens represents the R&D expenditures as a percentage of sales, while sales is the annual sales volume of the firm expressed in millions of dollars.

(b) ln̂(Q)= 3.717 – 1.21ln(P), where Q is the per capita consumption of chicken in pounds and P is the price of chicken in dollars.

Suppose that for years the mean of population 1 has been accepted as the same as the mean of population 2, but that now population 1 is believed to have a greater mean than population 2. Letting α = 0.05 and assuming the populations have equal variances and x is approximately normally distributed, use the following data to test this belief. Sample 1: 43.6, 45.2, 43.4, 49.1, 45.2, 45.6, 40.8, 46.5, 48.3, 45.6    Sample 2: 40.1, 36.0, 42.2, 42.3, 43.1, 38.8, 38.8, 43.3, 41.0, 40.8

(Round the intermediate values to 3 decimal places. Round your answer to 2 decimal places.)

Observed t = ?

Decision is to reject null hypothesis or fail to reject null hypothesis?

Win/Loss and With/Without Joe: Joe plays basketball for the Wildcats and missed some of the season due to an injury. The win/loss record with and without Joe is summarized in the contingency table below.

Observed Frequencies: O_i's

The Test: Test for a significant dependent relationship between the outcome of the game (win/lose) and whether or not Joe played. Conduct this test at the 0.01 significance level.

(a) What is the null hypothesis for this test?

H₀: The outcome of the game and whether or not Joe plays are dependent variables. H₀: The outcome of the game and whether or not Joe plays are independent variables.      H₀: The probability of winning is dependent upon whether or not Joe plays.

(b) What is the value of the test statistic? Round to 3 decimal places unless your software automatically rounds to 2 decimal places.

χ²

=

(c) Use software to get the P-value of the test statistic. Round to 4 decimal places unless your software automatically rounds to 3 decimal places.
P-value =

(d) What is the conclusion regarding the null hypothesis?

reject H₀ fail to reject H₀    

(e) Choose the appropriate concluding statement.

We have proven that Joe causes the team to do better. The evidence suggests that the outcome of the game is dependent upon whether or not Joe played.     There is not enough evidence to conclude that the outcome of the game is dependent upon whether or not Joe played. We have proven that the outcome of the game is independent of whether or not Joe played.

The assets​ (in billions of​ dollars) of the four wealthiest people in a particular country are 33, 30, 19, 12.

Assume that samples of size n=2 are randomly selected with replacement from this population of four values.

a. After identifying the 16 different possible samples and finding the mean of each​ sample, construct a table representing the sampling distribution of the sample mean. In the​ table, values of the sample mean that are the same have been combined.

​

A sample of n=12 students were given a test to see how quickly they could solve a task, they were then given feedback and tested again. On average, students completed the task 75 seconds faster with a standard deviation of 20 seconds. Did the feedback help improve task completion in terms of how quickly the task was completed? *We're using t tests, so I'd assume a two sample t test.*