Choose a point at random from the unit square [0, 1] × [0, 1]. We also choose the second random point, independent of the first, uniformly on the line segment between (0, 0) and (1, 0). The random variable A is the area of a triangle with its corners at (0, 0) and the two selected points. Find the probability density function (pdf) of A.

A random sample of 20 workers in a factory were asked to report the age of their car and how many miles the vehicle had on it. A computer printout resulted in the following information.

1. Find the LSRL
2. A new worker starts next week and we know that his car is 7 years old, how many miles would you expect to be on his car?
3. Interpret the slope of the LSRL in the context of this problem.
4. Find a 95% confidence interval for the slope of the LSRL
5. Without calculating it what do you know about an 85% confidence interval compared to your answer in part b?

Suppose the test scores for a college entrance exam are normally distributed with a mean of 450 and a s. d. of 100.
a. What is the probability that a student scores between 350 and 550?

b. If the upper 3% scholarship, what score must a student receive to get a scholarship?

c. Find the 60th percentile of the test scores.

d. Find the middle 30% of the test scores.

USE R AND SHOW CODES!!

1.a. There is a theorem that the ratios of blood types O, A, B and AB are 49:38:9:4,

respectively. A research team, investigating a small community and obtained the following frequencies of blood type.

Blood type

Blood Type O A B AB

Frequency 87 59 20 4

Test the hypothesis that the proportions in this community do not differ significantly

from those in the general theorem.

1.b. The severity of participants' migraines is clinically assessed. The following table

displays the severity classifications for patients assigned to the medical and the non

traditional therapies

Severity Classification

Therapy Minimal Moderate Severe

Medical 90 60 50

Non traditional 50 60 90

Is there any evidence that severity is independent of type of therapy?

Use the Chi-Square option in the Nonparametric Tests menu to answer the questions based on the following scenario. (Assume a level of significance of .05 and use information from the scenario to determine the expected frequencies for each category).

Scenario: During the analysis of the district data, it was determined that one high school had substantially higher Graduate Exit Exam scores than the state average and the averages of high schools in the surrounding districts. To better understand possible reasons for this difference, the superintendent conducted several analyses. One analysis examined the population of students who completed the exam. Specifically, the superintendent wanted to know if the distribution of special education, regular education, and gifted/talented test takers from the local high school differed from the statewide distribution. The obtained data are provided below. Description Special Education* Regular Education Gifted/Talented Number of students from the local high school who took the

*For purposes of testing, special education includes any student who received accommodations during the exam.

1. If the student distribution for the local high school did not differ from the state, what would be the expected percentage of students in each category?

2. What were the actual percentages of local high school students in each category? (Report final answer to two decimal places)

3. State an appropriate null hypothesis for this analysis.

4. What is the value of the chi-square statistic?

5. What are the reported degrees of freedom?

6. What is the reported level of significance?

7. Based on the results of the one-sample chi-square test, was the population of test taking students at the local high school statistically significantly different from the statewide population?

8. Present the results as they might appear in an article. This must include a table and narrative statement that reports and interprets the results of the analysis.

Note: The table must be created using your word processing program. Tables that are copied and pasted from SPSS are not acceptable.

Use the following table displaying 20 of the top (era-adjusted) grossing movies of all time, along with their Metacritic score (a weighted average of critics ratings), the amount of money they grossed (weighted for the year of release, in millions of dollars), and an indicator of whether or not the movie is a sequel to answer the following questions. You will need to use StatCrunch for this question.

a) Find the correlation between the Metacritic and Adj_Gross variables. Describe the strength and direction of the correlation.

b) Does having a higher Metacritic score have more an impact on gross revenue for original movies (non-sequels) or sequels? Put another way, does an increase of 1 in the Metacritic variable add more value to sequels or non-sequels? Justify your answer. You should be finding two different regression equations to answer this question.

c) Let’s say a new movie came out that received a Metacritic score of 89. Using your regression equations, how much money would it be expected to make if it is a sequel? How much money would it be expected to make if it is not a sequel?

The president of the American Insurance Institute wants to compare the yearly costs of auto insurance offered by two leading companies. He selects a sample of 15 families, some with only a single insured driver, others with several teenage drivers, and pays each family a stipend to contact the two companies and ask for a price quote. To make the data comparable, certain features, such as the deductible amount and limits of liability, are standardized. The data for the sample of families and their two insurance quotes are reported below.

At the 0.10 significance level, can we conclude that there is a difference in the amounts quoted? Hint: For the calculations, assume the "Midstates Car Insurance" as the first sample.

1. State the decision rule for 0.10 significance level: H₀: μ_d = 0 H₁: μ_d ≠ 0. (Negative values should be indicated by a minus sign. Round your answers to 3 decimal places.)

Reject H0 if T< _______ or t> _________

2. Compute the test statistic. (Negative value should be indicated by a minus sign. Round your answer to 2 decimal places.)

The test statistic is ___________

3. State your decision about the null hypothesis.

- Do not reject H0

- Reject H0

1. Suppose we have the following annual sales data for an automobile dealership:

Year                Sales                Trend

2009                121                     1

2010                187                     2

2011                165                     3

2012                134                     4

2013                155                     5

2014                167                     6

2015                200                     7

2016                206                     8

2017                221                    9

2018                231                 10

We want to forecast sales for 2019 and 2020 using either a simple trend model or a quadratic trend model. Use a within sample forecasting technique to determine the best model using the RMSE measure discussed in lecture. Once this model has been determined, provide actual forecasts for 2019 and 2020. Report the two RMSE values in your pdf or fax submission along with the actual forecasts. Submit your Excel file used to create these answers

Sonic Youth is playing at Nasa and Geir is leaving. Kim Gordon is on Bass. On the album Goo Kim plays 11 songs, the total is 49 minutes and 23 seconds. Kim also sings and sings on 6 tracks on the album. Sonic Youth have announced that they will be playing 21 songs at the concert. Assuming that the average length of a song is normally distributed and the standard deviation is 45 seconds. Hint: You have to give yourself one thing and another. Match units. What are the chances of Kim singing more than an hour at the concert?

A machine puts liquid into bottles of perfume. The amount of liquid put into each bottle, Dml, follows a normal distribution with mean 25ml.

Given that 15% of bottles contain less than 24.63ml

a.) find, to 2 decimal places, the value of k such that P(24.63<D<k) = 0.45

A random sample of 200 bottles is taken.

b.) Using a normal approximation, find the probability that fewer than half of these bottles contain between 24.63ml and k ml

The machine is adjusted so that the standard deviation of the liquid put in the bottles is now 0.16ml

Following the adjustments, Hannah believes that the mean amount of liquid put in each bottle is less than 25 ml

She takes a random sample of 20 bottles and finds the mean amount of liquid to be 24.94ml

c.) Test Hannah's belief at the 5% level of significance. You should state your hypothesis clearly.

Suppose we have the following values on the number of customers (X) and the average profits (Y) for fifteen stores:

Store                   Customers (X)          Average Profits (Y)

A                               161                              157

B                               99                              93

C                               135                              136

D                               120                              123

E                                164                              153

F                                221                              241

G                               179                              201

H                               204                              206

I                                 214                              229

J                                 101                              135

K                               231                              224

L                                206                              195

M                               248                              242

N                               107                              115

O                               205                              197

Use the Spearman’s Rank Correlation test at the 0.05 level to see if X and Y are significantly related.

The selling price of a new car is normally distributed with an average of $23090 and a standard deviation of $3060. a) What proportion of new cars will sell for more than $19130? probability = b) Assuming a normal distribution, within what selling prices will the middle 96% fall? lower = , upper =

1. You are excited because your new card game has arrived. This game consists of 40 unique cards. What is the number of ordered samples of 4 cards that can be drawn without replacement from the new deck of cards?

2. You are arranging flowers for a party. You have four white flowers, three yellow flowers, and two lavender flowers. How many different ways can these different colors be arranged?

3. You have a brown paper bag that contains two black balls and two red balls. Two balls are selected at random without replacement. You are told that at least one of the balls is red. What is the probability that the other ball you selected is also red? Give your answer to four decimal places.

4. You have a brown paper bag that contains 10 burnt orange balls and 10 navy blue balls (these are UVA’s colors where I went to graduate school, undergraduate was at UGA, thus I am a dawg). The balls are drawn at random, one at a time without replacement. Find the probability that the fourth burnt orange ball is the seventh ball drawn. Give your answer to four decimal places.

5 In a meadow there are 50 butterflies of which 10 have been tagged. An entomologist random catches 7 butterflies one at a time without replacement. What is the probability that exactly 2 tagged butterflies will be caught? Give your answer to four decimal places.

The Apparel Company makes expensive polo-style men's and women's short-sleeve knit shirts at its plant in Jamaica. The production process requires that material be cut into large patterned squares by operators, which are then sewn together at another stage of the process. If the squares are not of a correct length, the final shirt will be either too large or too small. In order to monitor the cutting process, management takes a sample of 4 squares of cloth every other hour and measures the length. The length of a cloth square should be 36 inches, and historically, the company has found the length to vary across an acceptable average of 2 inches.

Construct an R-chart for the cutting process using 3 sigma limits.

RAC has taken 10 additional samples with the following results:

Plot the new sample data on the control chart constructed above and comment on the process variability.

What might you have learned from this graph?

Construct an x-chart and an R-chart.

Plot the sample observations provided above.
Use both of the x- and R-charts to comment on the process.