. The joint probability density function of X and Y is given by

?(?, ?) = { ??^2? ?? 0 ≤ ? ≤ 2, 0 ≤ ?, ??? ? + ? ≤ 1

0 ??ℎ??????

(a) Determine the value of c.

(b) Find the marginal probability density function of X and Y.

(c) Compute ???(?, ?).

(d) Compute ???(?^2 + ?).

(e) Determine if X and Y are independent

You may need to use the appropriate appendix table or technology to answer this question.

Consider a multiple-choice examination with 50 questions. Each question has four possible answers. Assume that a student who has done the homework and attended lectures has a probability of 0.70 of answering any question correctly.

(a)

A student must answer 45 or more questions correctly to obtain a grade of A. What percentage of the students who have done their homework and attended lectures will obtain a grade of A on this multiple-choice examination? Use the normal approximation of the binomial distribution to answer this question.

(b)

A student who answers 34 to 39 questions correctly will receive a grade of C. What percentage of students who have done their homework and attended lectures will obtain a grade of C on this multiple-choice examination? Use the normal approximation of the binomial distribution to answer this question.

(c)

A student must answer 28 or more questions correctly to pass the examination. What percentage of the students who have done their homework and attended lectures will pass the examination? Use the normal approximation of the binomial distribution to answer this question.

(d)

Assume that a student has not attended class and has not done the homework for the course. Furthermore, assume that the student will simply guess at the answer to each question. What is the probability that this student will answer 28 or more questions correctly and pass the examination? Use the normal approximation of the binomial distribution to answer this question.

A professor surveyed a simple random sample of undergraduates at his large university. As part of this survey he asked about part time jobs the students had and the amount the students were paid per hour. The resulting data is given in the file “pay_survey.txt” This data is linked from the Moodle.

This data file contains the following variables.

Gender: Male or Female

Year: Year in school as Freshman, Sophomore, Junior or Senior

Pay: hourly rate of pay in dollars

We would like to determine if there is a significant difference between males and females in the hourly pay.

The post:

For this post you should

1. Use SAS to conduct a 2 sample t-test to determine if there is a significant difference between the genders in hourly pay.

Present the output from your SAS program. (You do not need to include the program or data for this post.)

Give an interpretation of the results in practical terms. Clearly explain the p-value and its meaning.

Present the assumptions of the tests you used with an explanation of their meaning in this situation. Also, present your conclusions about the assumptions.

Male Senior 16.82
Female Freshman 3.5
Female Sophomor 12
Female Junior 8
Female Sophomor 9.75
Female Junior 9.25
Female Sophomor 8
Female Junior 9.25
Male Junior 10
Female Junior 8.5
Female Junior 7.5
Female Sophomor 7.25
Female Senior 10
Female Sophomor 7.5
Male Sophomor 10
Female Freshman 8
Female Junior 8.75
Male Sophomor 12
Male Sophomor 7.25
Female Sophomor 15
Male Sophomor 8.25
Male Sophomor 8.5
Female Sophomor 8
Female Sophomor 8.5
Female Sophomor 9
Male Junior 10
Female Sophomor 7.25
Male Sophomor 9
Female Sophomor 7.5
Female Sophomor 8
Female Junior 8
Female Sophomor 8.5
Female Junior 10
Female Junior 9
Male Sophomor 10
Female Junior 8
Male Sophomor 8
Female Senior 9
Female Senior 7.25
Female Sophomor 7.25
Male Sophomor 7.25
Female Freshman 23
Female Sophomor 10
Female Sophomor 9
Female Sophomor 10
Female Sophomor 12
Female Senior 9
Female Junior 7.5
Female Junior 10
Female Junior 9
Female Junior 11
Female Junior 9.5
Female Sophomor 8.1
Female Senior 10
Female Junior 8.5
Female Sophomor 10
Male Freshman 8.5
Female Senior 21
Female Freshman 9
Female Freshman 12
Female Sophomor 10
Male Senior 10
Female Junior 20
Male Sophomor 8.5
Male Senior 8.5
Male Junior 8.25
Female Sophomor 10
Male Junior 9.5
Female Sophomor 9
Male Junior 7.25
Male Senior 9.25
Female Senior 10
Female Senior 11
Female Sophomor 9
Male Senior 10
Female Sophomor 7.5
Female Junior 12
Female Sophomor 12
Male Junior 14.25
Female Sophomor 7.25
Female Freshman 10
Female Junior 10
Female Sophomor 10
Female Junior 20
Female Senior 9
Male Sophomor 11
Female Junior 7.75
Male Sophomor 8
Female Freshman 7.5
Female Senior 8
Female Junior 8.75
Male Freshman 8
Female Sophomor 8.5
Female Sophomor 8.25
Male Junior 9.25
Male Junior 18.5
Female Sophomor 8
Female Sophomor 8
Female Junior 8.78
Male Sophomor 8
Male Junior 10.71
Female Sophomor 7.25
Female Junior 9.25
Male Sophomor 8.5
Female Sophomor 10
Female Senior 15.25
Female Freshman 7.25
Male Junior 8.5
Male Sophomor 8
Female Junior 15
Female Junior 12
Female Sophomor 10
Male Sophomor 14.5
Male Junior 11
Female Sophomor 7.25
Male Junior 10
Female Sophomor 8
Female Sophomor 8.25
Female Senior 14
Female Sophomor 9.25
Male Freshman 7.25
Female Sophomor 4.5
Female Sophomor 8
Male Sophomor 9
Female Sophomor 7.25
Female Sophomor 15
Female Freshman 12
Female Sophomor 10
Female Senior 7.25
Female Senior 10.25
Female Junior 8
Male Junior 10
Male Senior 16
Female Sophomor 7.25
Female Sophomor 10.25
Male Sophomor 18
Male Junior 9.25
Male Junior 15
Female Freshman 7.25
Female Freshman 8
Female Junior 10
Female Freshman 11
Male Senior 9
Male Senior 10.25
Female Senior 10

1) A friend of yours suggests that at least 10% of people at your university have had something stolen from them on campus in the past year. This is your population value. You take a random sample of 100 people in you classes to determine if your friend is correct, you think that they may be wrong. You find that 15% of them have had something stolen.

Would you have a one or two tailed test?

Two Tailed

One tailed

2) What is the strength of the following correlation? -0.512

Moderate

Strong

Weak

3) A friend of yours suggests that at least 10% of people at your university have had something stolen from them on campus in the past year. This is your population value. You take a random sample of 100 people in you classes to determine if your friend is correct, you think that they may be wrong. You find that 15% of them have had something stolen.

Would you use z or t to test this hypothesis?

Z Distribution

T Distribution

find and provide three examples of data visualizations which mislead a viewer (at least one should be in the business context).

include the screenshots/pictures of those visualizations. Explain why they mislead.

Advertising expenditures and sales for the last 5 quarters have been as follows:

In quarter 3, a new product was introduced that would influence sales in quarter 3. 4 and 5. The following model is established:

y=β₁x₁+β₂x₂+ ε

where y=sales, x₁=advertising expenditures, x₂ is a variable that is 1 when the new product is available and 0 otherwise, and ε is an error component. Find the least squares estimate of β_1.

The National Assessment of Educational Progress (NAEP) gave a test of basic arithmetic and the ability to apply it in everyday life to a sample of 840 men 21 to 25 years of age. Scores range from 0 to 500; for example, someone with a score of 325 can determine the price of a meal from a menu. The mean score for these 840 young men was x⎯⎯⎯x¯ = 272. We want to estimate the mean score μμ in the population of all young men. Consider the NAEP sample as an SRS from a Normal population with standard deviation σσ = 60.

(a) If we take many samples, the sample mean x⎯⎯⎯x¯ varies from sample to sample according to a Normal distribution with mean equal to the unknown mean score μμ in the population. What is the standard deviation of this sampling distribution?
(b) According to the 99.7 part of the 68-95-99.7 rule, 99.7% of all values of x⎯⎯⎯x¯ fall within _______ on either side of the unknown mean μμ. What is the missing number?
(c) What is the 99.7% confidence interval for the population mean score μμ based on this one sample? Note: Use the 68-95-99.7 rule to find the interval.

Now that you have completed this course in Statistics, please describe a concept covered in the course that you feel might be of assistance to you now or in the future. For example, using charts and graphs to graphically describe data at your job, or using one of the sampling methods discussed at the beginning of the course to generate sample data. Please be specific in explaining how you would use what you have learned in class to your benefit. (Marketing career example)

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Question: Two machines are used for filling plastic bottles with a net volume of 16.0 ounces. The filling p...

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Two machines are used for filling plastic bottles with a net volume of 16.0 ounces. The filling processes can be assumed to be normal. The quality engineering department is concerned that the second machine (Machine 2) is under filling the water bottles compared to the first machine (Machine 1). An experiment to address this concern is performed by taking a random sample from the output of each machine.

16.03   Machine 1
16.04   Machine 1
16.05   Machine 1
16.05   Machine 1
16.02   Machine 1
16.01   Machine 1
15.96   Machine 1
15.98   Machine 1
16.02   Machine 1
15.99   Machine 1
15.99   Machine 2
15.99   Machine 2
15.96   Machine 2
16   Machine 2
15.96   Machine 2
15.95   Machine 2
15.94   Machine 2
16.02   Machine 2
15.97   Machine 2
15.98   Machine 2

Q1

Compute the point estimate for the 95% confidence interval for the difference in mean volume of the bottles filled for Machine 1 versus Machine 2. (Round answer to 2 decimal places).

Q2

Using software or a statistical table, find the critical value for the 95% confidence interval for the difference in mean volume of the bottles filled for Machine 1 versus Machine 2. (Round answer to 2 decimal places).

Q3

Compute the standard error for the 95% confidence interval for the difference in mean volume of the bottles filled for Machine 1 versus Machine 2. (Round answer to 2 decimal places)

Suppose that the heights of male students at a university have a normal distribution with mean = 65 inches and standard deviation = 2.0 inches. A randomly sample of 10 students are selected to make up an intramural basket ball team.

i) What is the mean (mathematical expectation) of xbar?

ii) What is the standard deviation of x bar?

iii) What is the probability that the average height (x bar) of the team will exceed 69 inches?

iv) What is the probability that the average height (x bar) of the team will be between 62 and 70 inches?

This section of the assignment is designed to cultivate skills related to interpreting meaning from quantitative data. As part of a larger study, Speed and Gangestad (1997) collected ratings and nominations on a number of characteristics for 66 fraternity men from their fellow fraternity members. The following paragraph is taken from their results section:

. . . men's romantic popularity significantly correlated with several characteristics: best dressed (r = .48), most physically attractive (r = .47), most outgoing (r = .47), most self- confident (r = .44), best trendsetters (r =.38), funniest (r = .37), most satisfied (r = .32), and most independent (r =.28). Unexpectedly, however, men's potential for financial success did not significantly correlate with romantic popularity (r = .10). (p. 931)

Explain these results as if you were writing to a person who had never had a course in statistics. Specifically:

a) Explain what is meant by correlation coefficient using one of the correlations above as an example.

b) Provide your thoughts on the meaning of the pattern of results. (You could speculate on the meaning of the pattern of results, taking into account the issue of direct causality. You could also indicate what kinds of conclusions could NOT be drawn).  

Given their performance record and based on empirical rule what would be the upper bound of the range of sales values that contains 68% of the monthly sales?

A computer system uses passwords that contain exactly six characters, and each character is 1 of the 26 lowercase letters (a–z) or 26 uppercase letters (A–Z) or 10 integers (0–9). Let Ω denote the set of all possible passwords, and let A and B denote the events that consist of passwords with only letters or only integers, respectively. Determine the number of passwords in each of the following events.

(a) Password contains all lowercase letters given that it contains only letters (b) Password contains at least 1 uppercase letter given that it contains only letters (c) Password contains only even numbers given that it contains all numbers