In this exercise, we examine the effect of combining investments with positively correlated risks, negatively correlated risks, and uncorrelated risks. A firm is considering a portfolio of assets. The portfolio is comprised of two assets, which we will call ''A" and "B." Let X denote the annual rate of return from asset A in the following year, and let Y denote the annual rate of return from asset B in the following year. Suppose that E(X) = 0.15 and E(Y) = 0.20, SD(X) = 0.05 and SD(Y) = 0.06, and CORR(X, Y) = 0.30.

(a) What is the expected return of investing 50% of the portfolio in asset A and 50% of the portfolio in asset B? What is the standard deviation of this return?

(b) Replace CORR(X, Y) = 0.30 by CORR(X, Y) = 0.60 and answer the questions in part

(C). Do the same for CORR(X, Y) = -0.60, -0.30, and 0.0.

Let x represent the number of mountain climbers killed each year. The long-term variance of x is approximately σ2 = 136.2. Suppose that for the past 10 years, the variance has been s2 = 117.6. Use a 1% level of significance to test the claim that the recent variance for number of mountain-climber deaths is less than 136.2. Find a 90% confidence interval for the population variance. (a) What is the level of significance? State the null and alternate hypotheses. Ho: σ2 = 136.2; H1: σ2 < 136.2 Ho: σ2 = 136.2; H1: σ2 ≠ 136.2 Ho: σ2 = 136.2; H1: σ2 > 136.2 Ho: σ2 < 136.2; H1: σ2 = 136.2 (b) Find the value of the chi-square statistic for the sample. (Round your answer to two decimal places.) What are the degrees of freedom? What assumptions are you making about the original distribution? We assume a exponential population distribution. We assume a binomial population distribution. We assume a normal population distribution. We assume a uniform population distribution. (c) Find or estimate the P-value of the sample test statistic. P-value > 0.100 0.050 < P-value < 0.100 0.025 < P-value < 0.050 0.010 < P-value < 0.025 0.005 < P-value < 0.010 P-value < 0.005 (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Since the P-value > α, we fail to reject the null hypothesis. Since the P-value > α, we reject the null hypothesis. Since the P-value ≤ α, we reject the null hypothesis. Since the P-value ≤ α, we fail to reject the null hypothesis. (e) Interpret your conclusion in the context of the application. At the 1% level of significance, there is insufficient evidence to conclude that the variance for number of mountain climber deaths is less than 136.2 At the 1% level of significance, there is sufficient evidence to conclude that the variance for number of mountain climber deaths is less than 136.2 (f) Find the requested confidence interval for the population variance. (Round your answers to two decimal places.) lower limit upper limit Interpret the results in the context of the application. We are 90% confident that σ2 lies within this interval. We are 90% confident that σ2 lies below this interval. We are 90% confident that σ2 lies above this interval. We are 90% confident that σ2 lies outside this interval.

The types of raw materials used to construct stone tools found at an archaeological site are shown below. A random sample of 1486 stone tools were obtained from a current excavation site. Raw Material Regional Percent of Stone Tools Observed Number of Tools as Current excavation Site Basalt 61.3% 903 Obsidian 10.6% 169 Welded Tuff 11.4% 169 Pedernal chert 13.1% 196 Other 3.6% 49 Use a 1% level of significance to test the claim that the regional distribution of raw materials fits the distribution at the current excavation site. (a) What is the level of significance? State the null and alternate hypotheses. H0: The distributions are different. H1: The distributions are different. H0: The distributions are the same. H1: The distributions are the same. H0: The distributions are different. H1: The distributions are the same. H0: The distributions are the same. H1: The distributions are different. (b) Find the value of the chi-square statistic for the sample. (Round the expected frequencies to at least three decimal places. Round the test statistic to three decimal places.) Are all the expected frequencies greater than 5? Yes No What sampling distribution will you use? chi-square normal Student's t binomial uniform What are the degrees of freedom? (c) Find or estimate the P-value of the sample test statistic. P-value > 0.100 0.050 < P-value < 0.100 0.025 < P-value < 0.050 0.010 < P-value < 0.025 0.005 < P-value < 0.010 P-value < 0.005 (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis of independence? Since the P-value > α, we fail to reject the null hypothesis. Since the P-value > α, we reject the null hypothesis. Since the P-value ≤ α, we reject the null hypothesis. Since the P-value ≤ α, we fail to reject the null hypothesis. (e) Interpret your conclusion in the context of the application. At the 0.01 level of significance, the evidence is sufficient to conclude that the regional distribution of raw materials does not fit the distribution at the current excavation site. At the 0.01 level of significance, the evidence is insufficient to conclude that the regional distribution of raw materials does not fit the distribution at the current excavation

Assume that adults have IQ scores that are normally distributed with a mean of 102.6and a standard deviation of 18.7

Find the probability that a randomly selected adult has an IQ greater than 131.5

​(Hint: Draw a​ graph.)

The probability that a randomly selected adult from this group has an IQ greater than

131.5 is ????

In your rental shop, you know that on the busiest day you can expect 150 rentals. You also know that, historically, 60% of your customers rent skis and 40% rent snowboards.

a) If you decide that you only need to have 65 snowboards in stock, what is the probability that you will run out of snowboard rentals on any specific day?

      [ Choose ]            0.2259            0.1794            0.5310            0.0575            0.0664            0.5966            0.0409            0.0000      

b) What if you increase your supply to 70 showboards. Now, what is the probability that you will run out?

      [ Choose ]            0.2259            0.1794            0.5310            0.0575            0.0664            0.5966            0.0409            0.0000      

c) What is the probability that you will rent at most 90 skis on any given day?

      [ Choose ]            0.2259            0.1794            0.5310            0.0575            0.0664            0.5966            0.0409            0.0000      

d) What is the probability that you will rent exactly 90 skis on any given day

An article gave a scatter plot along with the least squares line of x = rainfall volume (m3) and y = runoff volume (m3) for a particular location. The accompanying values were read from the plot.

x 8 12 14 16 23 30 40 52 55 67 72 81 96 112 127

y 4 10 13 15 15 25 27 47 38 46 53 69 82 99 103

(a) Does a scatter plot of the data support the use of the simple linear regression model? Yes, the scatterplot shows a reasonable linear relationship. Yes, the scatterplot shows a random scattering with no pattern. No, the scatterplot shows a reasonable linear relationship. No, the scatterplot shows a random scattering with no pattern.

(b) Calculate point estimates of the slope and intercept of the population regression line. (Round your answers to five decimal places.) slope intercept

(c) Calculate a point estimate of the true average runoff volume when rainfall volume is 51. (Round your answer to four decimal places.) m3

(d) Calculate a point estimate of the standard deviation σ. (Round your answer to two decimal places.) m3

(e) What proportion of the observed variation in runoff volume can be attributed to the simple linear regression relationship between runoff and rainfall? (Round your answer to four decimal places.)

A random sample of 145 recent donations at a certain blood bank reveals that 81 were type A blood. Does this suggest that the actual percentage of type A donations differs from 40%, the percentage of the population having type A blood? Carry out a test of the appropriate hypotheses using a significance level of 0.01.
State the appropriate null and alternative hypotheses.

H₀: p = 0.40
H_a: p > 0.40 H₀: p = 0.40
H_a: p < 0.40     H₀: p = 0.40
H_a: p ≠ 0.40 H₀: p ≠ 0.40
H_a: p = 0.40

Calculate the test statistic and determine the P-value. (Round your test statistic to two decimal places and your P-value to four decimal places.)

State the conclusion in the problem context.

Reject the null hypothesis. There is not sufficient evidence to conclude that the percentage of type A donations differs from 40%. Do not reject the null hypothesis. There is sufficient evidence to conclude that the percentage of type A donations differs from 40%.     Reject the null hypothesis. There is sufficient evidence to conclude that the percentage of type A donations differs from 40%. Do not reject the null hypothesis. There is not sufficient evidence to conclude that the percentage of type A donations differs from 40%.

Would your conclusion have been different if a significance level of 0.05 had been used?

Yes No    

You may need to use the appropriate table in the Appendix of Tables to answer this question.

Preliminary data analyses indicate that you can consider the assumptions for using nonpooled​ t-procedures satisfied. Researchers randomly and independently selected 30 prisoners diagnosed with chronic posttraumatic stress disorder​ (PTSD) and 24 prisoners that were diagnosed with PTSD but had since recovered​ (remitted). The​ ages, in​ years, at arrest yielded the summary statistics shown in the table to the right. At the 10​% significance​ level, do the data provide sufficient evidence to conclude that a difference exists in the mean age at arrest of prisoners with chronic PTSD and remitted​ PTSD? Chronic Remitted x overbar 1 equals 26.9 x overbar 2 equals 22.4 s 1 equals 4 s 2 equals 8 n 1 equals 30 n 2 equals 24 What are the hypotheses for the nonpooled​ t-test? A. H0​: mu1equalsmu2 Ha​: mu1greater thanmu2 B. H0​: mu1greater than or equalsmu2 Ha​: mu1less thanmu2 C. H0​: mu1equalsmu2 Ha​: mu1not equalsmu2 D. H0​: mu1equalsmu2 Ha​: mu1less thanmu2 Find the test statistic. tequals nothing ​(Round to three decimal places as​ needed.) Find the​ P-value. Pequals nothing ​(Round to four decimal places as​ needed.) What is the conclusion of the hypothesis​ test? ▼ the null​ hypothesis, meaning that the data ▼ sufficient evidence to conclude that a difference exists in the mean age at arrest of prisoners with chronic PTSD and remitted PTSD.

The paint used to make lines on roads must reflect enough light to be clearly visible at night. Let μ denote the true average reflectometer reading for a new type of paint under consideration. A test of H₀: μ = 20 versus H_a: μ > 20 will be based on a random sample of size n from a normal population distribution. What conclusion is appropriate in each of the following situations? (Round your P-values to three decimal places.)

(a)    n = 19, t = 3.3, α = 0.05
P-value =

State the conclusion in the problem context.

Reject the null hypothesis. There is not sufficient evidence to conclude that the new paint has a reflectometer reading higher than 20. Do not reject the null hypothesis. There is not sufficient evidence to conclude that the new paint has a reflectometer reading higher than 20.     Reject the null hypothesis. There is sufficient evidence to conclude that the new paint has a reflectometer reading higher than 20. Do not reject the null hypothesis. There is sufficient evidence to conclude that the new paint has a reflectometer reading higher than 20.

(b)    n = 8, t = 1.7, α = 0.01
P-value =

State the conclusion in the problem context.

Reject the null hypothesis. There is not sufficient evidence to conclude that the new paint has a reflectometer reading higher than 20. Do not reject the null hypothesis. There is sufficient evidence to conclude that the new paint has a reflectometer reading higher than 20.     Do not reject the null hypothesis. There is not sufficient evidence to conclude that the new paint has a reflectometer reading higher than 20. Reject the null hypothesis. There is sufficient evidence to conclude that the new paint has a reflectometer reading higher than 20.

(c)    n = 25,

t = −0.3

P-value =

State the conclusion in the problem context.

Do not reject the null hypothesis. There is not sufficient evidence to conclude that the new paint has a reflectometer reading higher than 20. Reject the null hypothesis. There is not sufficient evidence to conclude that the new paint has a reflectometer reading higher than 20.     Do not reject the null hypothesis. There is sufficient evidence to conclude that the new paint has a reflectometer reading higher than 20. Reject the null hypothesis. There is sufficient evidence to conclude that the new paint has a reflectometer reading higher than 20.

You may need to use the appropriate table in the Appendix of Tables to answer this question.

Suppose that the length of an aeronautical component’s tensile strength test can be modeled as an exponential random variable with a mean of 4 years.

(i) What is the probability that the tensile strength exceeds the mean value by no more than 2 standard deviations?

(ii) Compute the median and the 64th percentile of the tensile strength.

29% of all college students major in STEM (Science, Technology, Engineering, and Math). If 33 college students are randomly selected, find the probability that

a. Exactly 9 of them major in STEM.  
b. At most 12 of them major in STEM.  
c. At least 8 of them major in STEM.  
d. Between 9 and 13 (including 9 and 13) of them major in STEM.

1. Answer the following questions regarding a standard 52-deck, which has 13 of each suite )heart, spades, clubs, and diamonds) and 3 face cards in each suit.

a. What is the probability that you will randomly receive an ace of diamonds or a queen of clubs?

b. If you were to draw a card, replace it, and shuffle, then what would be the probability of drawing the same card again?

c. Face cards are the King, Queen, or Jack cards for each suit. What is the probability of drawing a face card or a diamond card?

2. Dice have 6 sides with 1 through 6 dots on each side. There are 36 possible outcomes with the rolling of two dice. What is the probability that you could roll the sum of 7 with the pair of dice?

3. For a normal distribution with a mean of μ = 50 and σ = 10, find each probability value requested. Show your work in order to receive credit.

a. p(x>65)

b. p(x<47)

c. p(40 < X < 60)

the weight of a large number of miniature poodles are approximately normally distributed with a mean of 8 kilograms and a standard deviation of 0.9 Kg measurments are recorded to the nearest tenth of a kilogram.

a)draw a sketch of the distribution of poodle weights

b)what is the chance of a poodle weight over 9.8Kg?

c)what is the chance a poodle weight between 7.1 and 8.9 kilogram

d)75% of poodles weigh more than what weight

Background

This part is based on a study of premature mortality in Great Britain between 2012 and 2014. The dataset from this study includes information on premature mortality for 378 local authorities in Great Britain from 2012 to 2014. Premature mortality is measured as the number of individuals that die before the age of 70 in a cohort of 100,000. In addition to total premature mortality, the dataset also includes a breakdown by gender and socioeconomic indicators such as income, education and employment for each local authority.

Dataset

You can run the following line of code in R and this will load the data directly from the course website:

pmdata <- read.csv("https://uclspp.github.io/datasets/data/pmgb2012_2014.csv")

Codebook

The codebook describes the variables in the dataset.

Variable                            Description

code                                 Unique identifierer for each local authority

country                             1 = England, 2 = Scotland, 3 = Wales

pop_density                    1 = low, 2 = medium, 3 = high

pmdeaths_total              Number of premature deaths out of 100,00

pmdeaths_female          Number of premature deaths among women, out of 100,000

pmdeaths_male             Number of premature deaths among men, out of 100,000

mean_income                  Mean income in the local authority

edu_level3                       Qualification: proportion of the population with A level

edu_level4                        Qualification: proportion of the population with degree-level education or equivalent

2a. Descriptive Statistics

• Using the appropriate measures, report and interpret the central tendency and dispersion for the following variables:

– edu_level3

– edu_level4

– pop_density

2b. Visualization •

Produce a scatter plot of premature mortality (pmdeaths_total) on the y-axis and degree-level education (edu_level4 ) on the x-axis

• Provide an explanation of the substantive meaning of the graphs. What do they tell us about the association between premature mortality and levels of education in Great Britain?

• Produce a box plot that compares premature mortality in England, Scotland, Wales.

• What does the plot tell us about how premature mortality varies across the three countries?

2c. Difference in Means

• Calculate the mean difference between premature mortality among men and women in Great Britain.

• Conduct t-test to establish whether the difference between the premature mortality of men and women is statistically significant at the 95% confidence level.

• Interpret the results of the t-test both statistically and substantively

• Interpret the confidence interval of the difference in means

2d. Linear Regression

• Estimate a linear regression model to analyse the relationship between mean income and premature mortality in each local authority. The dependent variable is pmdeaths_total and the independent variable is mean_income

• Present a table with the output of the regression model

• Interpret the main coefficient of interest (mean_income)

• Interpret the estimated intercept term of the regression model

• Interpret the R2 term of the regression model

Weight^ (pounds) = -150 + (5 * Inches Tall)

a)Interpret the intercept (bo)in the context of the problem.

b)Interpret the slope (b1)in the context of the problem.

c)Predict the weight of a person who is 6 feet tallusing the regression line.

d)If a person 6 feet tall truly weighs 200 pounds, what is their residual value?

e)Give 1 example of an outlier that would not be an influential value and 1 example of an outlier that would be, in the context of this height and weight example.