An urn contains n white balls and m black balls. ( m and n are both positive numbers.)

(a) If two balls are drawn without replacement , what is the probability that both balls are the same color?

(b) If two balls are drawn with replacement (i.e., One ball is drawn and it’s color recorded and then put back. Then the second ball is drawn.) What is the probability that both balls are the same color.

(c) Show that the probability in part (b) is always larger than the one in part (a)

Please show in EXCEL how to express with function formula.

As we discussed during Week 3, researchers decide to ask 4 tie-purchasing customers whether they bought a bow tie or normal tie. According to national data, 3% of all ties purchased are bow ties.

Not surprisingly, this problem can be easily framed as a Binomial Distribution problem, as it meets all of the conditions outlined in our Week 4 (Monday) class. Further, we can identify the purchase of a bow tie with fixed probability of .03 as a success; the purchase of a standard tie with fixed probability of .97 as a failure. In other words:

p = .03, q=(1-p) = .97, n=4.

1.         On your Excel sheet, type in “PDF by hand” in cell A39. Then in cells A40 through A 44, use the COMBIN and POWER (or “^”) functions to compute the probability of x = 0, 1, 2, 3, 4 in each of the cells. In other words, in cell A40, compute P(X=0); cell A41, P(X=1); and so on. (That is, you’re using Excel to apply the binomial formula shown in class.)

              Hint: For P(X=0), enter the following:

“=COMBIN(4,0)*POWER(0.03,0)*POWER(1-0.03,4-0)” or

             

=COMBIN(4,0)*.03^0* (1-.03)^(4-0)”

(Note: for this problem, and all others in this Exercise, please round the numbers to 4 digits using Excel. You can do this after the fact from the toolbar, as we’ve shown before in class.)

Finally, in cell A46, use the SUM function to sum the individual probabilities you’ve computed in (a).

2.         In cell A48, compute the probability that at least one of the four persons purchased a bow tie by adding up each of the relevant probabilities, rounding to 4 digits.   (Note: do not type in any values. To receive credit, you must reference the relevant cells in your equations.)

              Then, in cell A49, compute the probability that at least one of the four persons purchased a bow tie by taking the complement of the event, X = 0. (Note: again, do not type in any specific values. Reference the relevant cells from the values you’ve already computed.) Round to 4 digits. In cell B49, type in “1-P(X=0)”.

3.         In cells B40:B44, compute the cumulative probabilities for x = 0, 1, 2, 3, 4, respectively, by entering appropriate commands that reference your computed pdf probabilities in cells A40 through A44. Again, round to 4 digits. Label the column “CDF by hand” in cell B39.

              Then, in cells, C40, C41, …, C44 type in “x=0, x=1, x=2, x=3, x=4”, respectively.

4.         Now, in cell D39 type PDF BINOM. Now use the BINOM.DIST function to populate cells D40:D44 with the PDFs of x = 0, 1, 2, 3, 4. (Remember, the PDF requires the “FALSE” entry.) Make sure you round your answers to four digits.

5.         Finally, using BINOM.DIST, populate cells E40:E44 with the CDFs of x = 0, 1, 2, 3, 4, rounding your answers to four digits. (Remember, the CDF requires the BINOM.DIST “TRUE” entry. Further, you must use BINOM.DIST, i.e., do not reference the output found in any other column.) As in (d), label the column by typing “CDF BINOM” in cell E39.

              In cell E46, reference via an “=” sign the appropriate cell that identifies the probability that fewer than 2 persons purchased a bow tie. In cell F46 type in “P(X≤1)”.

As we discussed during Week 3, researchers decide to ask 4 tie-purchasing customers whether they bought a bow tie or normal tie. According to national data, 3% of all ties purchased are bow ties.

Not surprisingly, this problem can be easily framed as a Binomial Distribution problem, as it meets all of the conditions outlined in our Week 4 (Monday) class. Further, we can identify the purchase of a bow tie with fixed probability of .03 as a success; the purchase of a standard tie with fixed probability of .97 as a failure. In other words:

p = .03, q=(1-p) = .97, n=4.

1.         On your Excel sheet, type in “PDF by hand” in cell A39. Then in cells A40 through A 44, use the COMBIN and POWER (or “^”) functions to compute the probability of x = 0, 1, 2, 3, 4 in each of the cells. In other words, in cell A40, compute P(X=0); cell A41, P(X=1); and so on. (That is, you’re using Excel to apply the binomial formula shown in class.)

              Hint: For P(X=0), enter the following:

“=COMBIN(4,0)*POWER(0.03,0)*POWER(1-0.03,4-0)” or

             

=COMBIN(4,0)*.03^0* (1-.03)^(4-0)”

(Note: for this problem, and all others in this Exercise, please round the numbers to 4 digits using Excel. You can do this after the fact from the toolbar, as we’ve shown before in class.)

Finally, in cell A46, use the SUM function to sum the individual probabilities you’ve computed in (a).

2.         In cell A48, compute the probability that at least one of the four persons purchased a bow tie by adding up each of the relevant probabilities, rounding to 4 digits.   (Note: do not type in any values. To receive credit, you must reference the relevant cells in your equations.)

              Then, in cell A49, compute the probability that at least one of the four persons purchased a bow tie by taking the complement of the event, X = 0. (Note: again, do not type in any specific values. Reference the relevant cells from the values you’ve already computed.) Round to 4 digits. In cell B49, type in “1-P(X=0)”.

3.         In cells B40:B44, compute the cumulative probabilities for x = 0, 1, 2, 3, 4, respectively, by entering appropriate commands that reference your computed pdf probabilities in cells A40 through A44. Again, round to 4 digits. Label the column “CDF by hand” in cell B39.

              Then, in cells, C40, C41, …, C44 type in “x=0, x=1, x=2, x=3, x=4”, respectively.

4.         Now, in cell D39 type PDF BINOM. Now use the BINOM.DIST function to populate cells D40:D44 with the PDFs of x = 0, 1, 2, 3, 4. (Remember, the PDF requires the “FALSE” entry.) Make sure you round your answers to four digits.

5.         Finally, using BINOM.DIST, populate cells E40:E44 with the CDFs of x = 0, 1, 2, 3, 4, rounding your answers to four digits. (Remember, the CDF requires the BINOM.DIST “TRUE” entry. Further, you must use BINOM.DIST, i.e., do not reference the output found in any other column.) As in (d), label the column by typing “CDF BINOM” in cell E39.

              In cell E46, reference via an “=” sign the appropriate cell that identifies the probability that fewer than 2 persons purchased a bow tie. In cell F46 type in “P(X≤1)”.

Your job is to fully staff your facility at the lowest cost. The facility must have at least 2 people w working from 6am-8pm Monday thru Friday and at least 1 person working from 10am to 6 pm on Saturdays. Nobody works on Sundays. Full time staff must work 8 hours a day, five days a week. Part time staff work 4 hours a day, 5 days a week. Nobody is allowed to work overtime. All employees receive the same hourly rate. What is the number of part time and full time employees are needed in order to fully staff the operation with the smallest total labor cost?

FT employees needed?:

PT employees needed?:

The probability the adult used tobacco products is 0.300.

The probability the adult binge drank alcohol is 0.384.

The probability the adult drank any alcohol is 0.571.

The probability the adult used tobacco products and did not drink any alcohol is 0.174.

a) Given that the adult drank alcohol, what is the probability the adult binge drank alcohol?

b) What is the probability the adult used tobacco products and drank any alcohol?

c) What is the probability the adult did not use tobacco products and did not drink any alcohol?

Choose ONE of the random variables from the options provided in each part. (a) Confirm the essential properties of the probability function for the binomial or Poisson or geometric random variable. [5 marks] (b) Derive the mean of the binomial or Poisson or geometric random variable from first principles (i.e. using the probability function and the definition of expectation). [7 marks] (c) Confirm the essential properties of the probability density function for the uniform or exponential random variable. [5 marks] (d) Derive the cumulative distribution function for the uniform or exponential random variable. Show that this function meets the necessary requirements for such a function (state what these are, and show that they are met). [8 marks] (e) Derive P(x1 < X < x2) for the uniform or exponential random variable. Your answer should be a function of x1, x2 and the parameters of the distribution you choose. You may use your result in (d), but if you choose a different random variable, you must start from f(x). [5 marks] (f) Derive the mean of the uniform or exponential or normal random variable by any method. [5 marks] (g) Derive the moment generating function for any one of the random variables listed in the test resource. [6 marks]Derive E(X2 ) for any one of the random variables listed in the test resource, from first principles (i.e. using f(x) and the definition of expectation) or by using its moment generating function. Hence or otherwise, derive the variance of that random variable. You may assume linearity of expectation as it app

1) You buy a lottery ticket to a lottery that costs $10 per ticket. There are only 100 tickets available to be sold in this lottery. In this lottery there are one $500 prize, two $100 prizes, and four $25 prizes.

Show its probability distribution in the form of a table.

What is the standard deviation of your gain or loss?

What type of skewness does the probability distribution represent?

|    | a1   | a2   |
|----|------|------|
| b1 | 0.37 | 0.16 |
| b2 | 0.23 | ?    |

1. What is ?(?=?2,?=?2)P(A=a2,B=b2)?

2. Observing events from this probability distribution, what is the probability of seeing (a1, b1) then (a2, b2)?

3. Calculate the marginal probability distribution, ?(?)P(A).

4. Calculate the marginal probability distribution, ?(?)P(B).

-Event time T follows an exponential distribution with a mean of 40
-Censoring time Tc follows an exponential distribution with a mean of 25
-Generate 500 observations, with censoring flag indicating whether censoring happened before events

Question: What do you think the percent of censoring should be? Show your calculation or reasoning.

There is one 1$ bill and one 5$ bill in your left pocket and three 1$ bills in your right pocket. You move one bill from the left pocket to the right pocket. After that you take one the remaining bill from the left pocket and one of the bills at random from your right pocket. Let ? denote the amount of money that you take from the left pocket and ? denote the amount of money that you take from the right pocket.

(a) Covariance between ? and ? is...

(b) Let ? denote the total amount of money that you get from your pockets. V??(?) is...

(c) Let ? denote the share of money that you get from the left pocket, i.e. ? ? . Calculate the mean of U.

Random sample of size n=19 is taken from a Normal population, sample mean is 11.5895, standard deviation is 1.0883.

1. At the 2% level, test whether it is reasonable to believe that the true population variance is larger than 1.

Using the scenario from above, do the following

2. Derive the power function. Show your work.
3. Using R, graph the power function for 0.5 < sigma^2 < 3.5.
4. Pretend that the sample size was actually 56. Plot this power function on the same graph.
5. Based on the power functions graphed in c, why is the test described in part c "better" than the original test? Explain your answer using the power functions.

For PART B USE A TI 84 AND LIST THE STEPS THAT YOU DID! THANK YOU

Researchers want to determine whether all bags of Skittles® have the same proportion of colors regardless of the flavor of Skittles®. To test this, they randomly sampled king-size bags of each flavor and recorded their findings in the table.

Flavor	Skittles® Color
	Red	Orange	Yellow	Blue	Green
Original	15	20	18	12	16
Tropical	10	7	9	18	5
Wild Berry	16	12	13	8	10

Part A: What are the correct degrees of freedom for this table? (2 points)

Part B: Calculate the expected count for the number of blue tropical Skittles®. Show your work. (3 points)

Part C: Is there sufficient evidence that there is a difference in the proportion of colors for the different flavors of Skittles®? Provide a statistical justification for your conclusion. (5 points)

The mean cost of domestic airfares in the United States rose to an all-time high of $400 per ticket. Airfares were based on the total ticket value, which consisted of the price charges by the airlines plus any additional taxes and fees. Assume domestic airfares are normally distributed with a standard deviation of $120. Use Table 1 in Appendix B.

a. What is the probability that a domestic airfare is $555 or more (to 4 decimals)?

b. What is the probability that a domestic airfare is $250 or less (to 4 decimals)?

c. What if the probability that a domestic airfare is between $300 and $470 (to 4 decimals)?

d. What is the cost for the 4% highest domestic airfares? [(round to nearest dollar) (more or less)]

2) A random sample of 10 miniature Tootsie Rolls was taken from a bag. Each piece was weighed on a very accurate scale. The results in grams were:

3.087 3.131 3.241 3.241 3.270 3.353 3.400 3.411 3.437 3.477

-Assuming a normal population, use Minitab to construct a 95 percent confidence interval for the true mean weight. You will need to enter the data. Attach or include your output. (4 points)

-Write a sentence using the confidence interval found in part a. (3 points)

-Use Minitab to construct a histogram of the sample data. Use the histogram to determine if the

assumption of normality is a valid assumption. State your findings. (4 points)

-What sample size would be necessary to estimate the true weight with an error of ± 0.025 gram

with 90 percent confidence? (4 points)

Explain the differences between a sample and a population, and offer example (s) to further illustrate your explanation

Answer the following questions and use Excel or this document to show your work.

1. Consider the following results for two samples randomly taken from two normal populations with equal variances.

	Sample I	Sample II
Sample Size	28	35
Sample Mean	48	44
Population Standard Deviation	9	10

a. Develop a 95% confidence interval for the difference between the two population means.

b. Is there conclusive evidence that one population has a larger mean? Explain.