A college looked at a random sample of students who worked the previous summer.  

Use the following information to answer questions 1 - 7.

The following data show the average earnings and sample standard deviations for a random sample of males and a random sample of females.

The following degrees of freedom may be helpful: 25.49

1) Find the point estimate of the difference between the mean summer earnings for males and the mean summer earnings for females.

2) Find the 95% confidence interval for the difference between the means of the two populations.

3) The college is interested in showing that the mean summer earnings for the males is greater than the mean summer earnings for females. What null and alternate hypothesis should the college use?

4) The college is interested in showing that the mean summer earnings for males is greater than the mean summer earnings for females. What is the value of the test statistic?
a) 3.17
b) -3.17
c) 2.10
d) -2.10
e) 1.16
f) -1.16
g) none of the above

5) The null hypothesis in problem 3 is to be tested at the 5% level of significance. The rejection region (regions) from the table is (are):
a)           z≤-1.96 or z≥1.96
b)           z≤1.96
c)            z≤-1.96
d)            z≤-1.645 or z≥1.645
e)            t≥1.708
f)            t≤-1.708
g)            t≤-1.708or t≥1.708
h)            t≤-2.060
i)             t≥2.060
j)             t≤-2.060 or t≥2.060

6) If the null hypothesis is tested in problem 4 at the 5% level, the null hypothesis should be:
a) rejected
b) not rejected
c) impossible to determine

7) Find the p-value for the hypothesis test in problem 4.
a) p-value>.10
b) .05 c) .025 d) .01 e) .005 f) p-value<.005

Steps on how to get the answers would be appreciated. Thanks.

1. (a) A statistician randomly sampled 100 observations and found

= 106 and s = 35. Calculate the t-statistic and p-value for testing

H₀: μ = 100 vs H_A: μ > 100.

Carry out the test at the 1% level of significance.

(b) Repeat part (a), with s = 25.

(c) Repeat part (a), with s = 15.

(d) Discuss what happens to the t-statistic and the p-value when the standard deviation decreases.

2. Repeat Question 1 using H_A: μ ≠ 100.

What are the advantages and disadvantages of strip mines vs. underground mines?

The number of chocolate chips in a bag of chocolate chip cookies is approximately normally distributed with a mean of 1262 chips and a standard deviation of 118 chips. ​(a) Determine the 26th percentile for the number of chocolate chips in a bag. ​(b) Determine the number of chocolate chips in a bag that make up the middle 98​% of bags. ​(c) What is the interquartile range of the number of chocolate chips in a bag of chocolate chip​ cookies?

Show work please. So i can understand. Also how do you use and find the standard normal table or x table?

1. Have you ever played rock-paper-scissors (or Rochambeau)? It’s considered a “fair game” in that the two players are equally likely to win (like a coin toss). Both players simultaneously display one of three hand gestures (rock, paper, or scissors), and the objective is to display a gesture that defeats that of your opponent. The main gist is that rocks break scissors, scissors cut paper, and paper covers rock. We investigated some results of the game rock-paper-scissors, where the researchers had 119 people play rock-paper-scissors against a computer. They found 66 players (55.5%) started with rock, 39 (32.8%) started with paper, and 14 (11.8%) started with scissors.We want to see if players start with scissors with a probability that is different from 1/3.

a) Identify the parameter in the question. (hint: the long-run proportion of …) (1 pts)

b) State the null hypothesis in words. (1 pts)

The null would be a player that starts with scissors is 33%

c) State the null hypothesis using symbols. (0.5 pts)

H_O π=33%

d) State the alternative hypothesis in words. (1 pts)

The alternative would be a player that has scissors wouldn’t be 33%

e) State the alternative hypothesis in symbols. (0.5 pts)

H_a  π≠33%

f) What is the value for the statistics? (1 pts) Assign a symbol to this value. (0.5 pts)

z=-4.9

g) Using an “one proportion” applet, find the p-value. Report your p-value here: ___0____

(1 pts)

h) Based on this p-value, do we have strong evidence against the null hypothesis?  (1 pts)

Yes.

i) Summarize the conclusion in the context of the problem. (1 pts)

Lemington’sis trying to determine how many Jean Hudson dresses to order for the spring season. Demand for the dresses is assumed to follow a normal distribution with mean of 400 and standard deviation of 100. The contract between Jean Hudson and Lemington’s works as follows. At the beginning of the season, Lemington’s reserves x units of capacity. Lemington’s must take delivery for at least 0.8x dresses and can, if desired, take delivery on up to x dresses. Each dress sells for £160 and Hudson charges £50 per dress. If Lemington’s does not take delivery on all x dresses, it owes Hudson a £5 penalty for each unit of reserved capacity that is unused. For example, if Lemington’s orders 450 dresses and demand is for 400 dresses, Lemington’s will receive 400 dresses and owe Jean 400(£50) + 50(£5). How many units of capacity should Lemington’s reserve to maximise its expected profit? a) Set up a simulation model to help the company make the decision of how many units of capacity to reserve. b) Discuss on the obtained results and make suggestions. For example, what if the demand for the dress has different distributions?

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.