We have three light bulbs with lifetimes T1,T2,T3 distributed according to Exponential(λ1), Exponential(λ2), Exponential(λ3). In other word, for example bulb #1 will break at a random time T1, where the distribution of this time T1 is Exponential(λ1). The three bulbs break independently of each other. The three light bulbs are arranged in series, one after the other, along a circuit—this means that as soon as one or more light bulbs fail, the circuit will break. Let T be the lifetime of the circuit—that is, the time until the circuit breaks.

(a) What is the CDF of T, the lifetime of the circuit?

(b) Next, suppose that we only check on the circuit once every second (assume the times T1,T2,T3,T are measured in seconds). Let S be the first time we check the circuit and see that it’s broken. For example, if the circuit breaks after 3.55 seconds, we will only observe this when 4 seconds have passed, and so S = 4. Calculate the PMF of S.

(c) Finally, suppose that instead of checking on the circuit every second, we instead do the following: after each second, we randomly decide whether to check on the circuit or not. With probability p we check, and with probability 1−p we do not check. This decision is made independently at each time. Now let N be the number of times we check and see the circuit working. For example, if the circuit breaks at time 3.55, and our choices were to check at time 1 second, not to check at times 2 or 3 or 4, and to check at time 5, then N = 1, since the circuit was broken the 2nd time we checked. What is the PMF of N? (Hint: start by finding the joint PMF of N and S. It’s fine if your answer is in summation form.)

A nutritionist is interested in the relationship between cholesterol and diet. The nutritionist developed a non-vegetarian and vegetarian diet to reduce cholesterol levels. The nutritionist then obtained a sample of clients for which half are told to eat the new non-vegetarian diet and the other half to eat the vegetarian diet for three months. The nutritionist hypothesizes that the non-vegetarian diet will reduce cholesterol levels more. What can the nutritionist conclude with α = 0.01. Below are the cholesterol levels of all the participants after three months.

a) What is the appropriate test statistic?
---Select--- na z-test One-Sample t-test Independent-Samples t-test Related-Samples t-test

b)
Condition 1:
---Select--- cholesterol level non-vegetarian months diet vegetarian
Condition 2:
---Select--- cholesterol level non-vegetarian months diet vegetarian

c) Compute the appropriate test statistic(s) to make a decision about H₀.
(Hint: Make sure to write down the null and alternative hypotheses to help solve the problem.)
critical value =  ; test statistic =
Decision:  ---Select--- Reject H0 Fail to reject H0

d) If appropriate, compute the CI. If not appropriate, input "na" for both spaces below.
[  ,  ]

e) Compute the corresponding effect size(s) and indicate magnitude(s).
If not appropriate, input and/or select "na" below.
d =  ;  ---Select--- na trivial effect small effect medium effect large effect
r² =  ;  ---Select--- na trivial effect small effect medium effect large effect

f) Make an interpretation based on the results.

Non-vegetarians had significantly higher cholesterol levels than vegetarians.

Non-vegetarians had significantly lower cholesterol levels than vegetarians.    

There was no significant cholesterol difference between non-vegetarians and vegetarians.

Prove "The Birthday Problem" in this regard,

Suppose there are some number of people in a room and we need need to consider all possible pairwise combinations of those people to compare their birthdays and look for matches.Prove the probability of the matches.

In this problem, we use your critical values table to explore the significance of r based on different sample sizes. (a) Is a sample correlation coefficient ρ = 0.82 significant at the α = 0.01 level based on a sample size of n = 3 data pairs? What about n = 14 data pairs? (Select all that apply.) No, because the absolute value of the given correlation coefficient is smaller than that for a sample size of n = 14 and α = 0.01. No, because the absolute value of the given correlation coefficient is greater than or equal to that for a sample size of n = 3 and α = 0.01. Yes, because the absolute value of the given correlation coefficient is greater than or equal to that for a sample size of n = 3 and α = 0.01. Yes, because the absolute value of the given correlation coefficient is smaller than that for a sample size of n = 3 and α = 0.01. Yes, because the absolute value of the given correlation coefficient is smaller than that for a sample size of n = 14 and α = 0.01. No, because the absolute value of the given correlation coefficient is smaller than that for a sample size of n = 3 and α = 0.01. Yes, because the absolute value of the given correlation coefficient is greater than or equal to that for a sample size of n = 14 and α = 0.01. No, because the absolute value of the given correlation coefficient is greater than or equal to that for a sample size of n = 14 and α = 0.01. Incorrect: Your answer is incorrect. (b) Is a sample correlation coefficient ρ = 0.42 significant at the α = 0.05 level based on a sample size of n = 18 data pairs? What about n = 26 data pairs? (Select all that apply.) Yes, because the absolute value of the given correlation coefficient is greater than or equal to that for a sample size of n = 26 and α = 0.05. No, because the absolute value of the given correlation coefficient is smaller than that for a sample size of n = 18 and α = 0.05. Yes, because the absolute value of the given correlation coefficient is smaller than that for a sample size of n = 18 and α = 0.05. Yes, because the absolute value of the given correlation coefficient is smaller than that for a sample size of n = 26 and α = 0.05. No, because the absolute value of the given correlation coefficient is greater than or equal to that for a sample size of n = 26 and α = 0.05. Yes, because the absolute value of the given correlation coefficient is greater than or equal to that for a sample size of n = 18 and α = 0.05. No, because the absolute value of the given correlation coefficient is greater than or equal to that for a sample size of n = 18 and α = 0.05. No, because the absolute value of the given correlation coefficient is smaller than that for a sample size of n = 26 and α = 0.05. Incorrect: Your answer is incorrect. (c) Is it true that in order to be significant, a ρ value must be larger than 0.90? larger than 0.70? larger than 0.50? What does sample size have to do with the significance of ρ? Explain your answer. No, a larger sample size means that a smaller absolute value of the correlation coefficient might be significant. No, sample size has no bearing on whether or not the correlation coefficient might be significant. Yes, a larger correlation coefficient of 0.70 means that the data will be significant. Yes, a larger correlation coefficient of 0.90 means that the data will be significant. Yes, a larger correlation coefficient of 0.50 means that the data will be significant.

Book :Business Analytics 6^th edition (data analysis and decision making)

By s. Chirstian albright and wayne. L . w

A

If your company makes a particular decision in the face of uncertainty, you estimate that it will either gain $10,000, gain $1000, or lose $5000, with probabilities 0.40, 0.30, and 0.30, respectively. You (correctly) calculate the EMV as $2800. However, you distrust the use of this EMV for decision-making purposes. After all, you reason that you will never receive $2800; you will receive $10,000, $1000, or lose $5000. Discuss this reasoning.

B

In the previous question, suppose you have the option of receiving a check for $2700 instead of making the risky decision described. Would you make the risky decision, where you could lose $5000, or would you take the sure $2700? What would influence your decision?

C

A potentially huge hurricane is forming in the Caribbean, and there is some chance that it might make a direct hit on Hilton Head Island, South Carolina, where you are in charge of emergency preparedness. You have made plans for evacuating everyone from the island, but such an evacuation is obviously costly and upsetting for all involved, so the decision to evacuate shouldn’t be made lightly. Discuss how you would make such a decision. Is EMV a relevant concept in this situation? How would you evaluate the consequences of uncertain outcomes

When using the test for homogeneity of variance, basically four outcomes or options can be considered if a violation of the homogeneity of variance assumption is violated. These include:

Option 1: Since a violation of the homogeneity of variance assumption occurred, the independent t-test may not be the appropriate statistical procedure for analysis of data. Therefore, we may opt for a lower order non-parametric statistical test.

Option 2: Since a violation of the homogeneity of variance assumption occurred, the independent t-test may not be the appropriate statistical procedure for analysis of data. Therefore, we will abandon the test all together.

Option 3: Even though we violated the homogeneity of variance assumption, we will continue to use the parametric measure due to the robust nature of the tests.

Option 4: Resample with a larger sample size and retest.

Provide MULTIPLE pros and cons of each of the four options and which one you think is the most valid.

Please answer ALL the questions (as there are multiple) embedded in the above task.

Q 1 . ( 8 marks) Answer the following:

a) Describe the difference between a discrete and a continuous ra ndom variable. Give an example of each.

b) Under what conditions might you choose to use a dot plot rather than a histogram?

c) Differentiate between retrospective and observational studies

d) What is the significance of the Bayes’ Theorem?   

For a poisson distribution where X ~ Pois(u), solve the questions below. Please show all work and all steps.

a.) Show that the pmf is a pmf using the criteria for verifying pmf (2 conditions).

b.) Show that E(X) = u

c.) Show that Var(X) = u

At Acme Bank the total amount of money that customers withdraw from an automatic teller machine (ATM) each day is believed to be normally distributed with a mean of $8600 and a variance of 6250000. i) At the beginning of weekday (M-F), the Acme Bank puts $10000 into the automatic teller machine. What is the probability that the ATM becomes empty before the end of the day. ii) How much should the Acme Bank put in the ATM each day in order to satisfy daily demand 99% of the time. iii) On a Saturday if customers withdraw from the (ATM) becomes below average, what is the probability that it will be below $6000? iv) If you select random samples of 25 from Acme Bank ATM users, what percentage of the sample means withdraws would be more than 10,000 from the ATM? v) In a random sample of size 25, what is the probability that sample mean withdraw exceeds 12000?

A numerical taxonomist studying the branched lamentous alga, Cladophora glom-
erata, believes she has discovered a new species. C. glomerata has a median branching
angle of 47. While the new species is quite similar to C. glomerata, she believes
the branching angle is less than that of C. glomerata. Measuring 17 specimens she
collected, she found the branching angles below. Are the angles of her sample
specimens significantly smaller than those of C. glomerata?
42 40 39 45 38 39 41 41 40
45 49 43 42 40 35 40 47

(c) Determine the 95 percent confidence interval for the population mean

(c) Evaluate the specified hypothesis by a t test, sign test and Wilcoxon signed-rank test

What is the minimum sample size required for estimating μ for N(μ, σ²=121) to within ± 2 with confidence levels 95%, 90%, and 80%?

a. Find the minimum sample size for the 95% confidence level. Enter an integer below. You must round up.

b. Find the minimum sample size for the 90% confidence level.

c. Find the minimum sample size for the 80% confidence level.

The Wall Street Journal’s Shareholder Scoreboard tracks the performance of 1000 major U.S. companies. The performance of each company is rated based on the annual total return, including stock price changes and the reinvestment of dividends. Ratings are assigned by dividing all 1000 companies into five groups from a (top 20%), b (next 20%), to e (bottom 20%). Shown here are the one-year ratings for a sample of 60 of the largest companies.

A) The study wants to test whether the largest companies differ in performance from the performance of the 1000 companies in the Shareholder Scoreboard. Clearly state the null and alternative hypotheses.

B) Compute the test statistic.

Please copy your R code and the result and paste them here.

C) At 5% significance level, compute the critical value for the test statistic and the p value for the test. Draw your conclusion.

Please copy your R code and the result and paste them here.

D) Use the function chisq.test() in R to run the test directly to confirm your results above are correct.

Please copy your R code and the result and paste them here.

What is the relationship between the number of minutes per day a woman spends talking on the phone and the woman's weight? The time on the phone and weight for 8 women are shown in the table below.

1. Find the correlation coefficient: r=r=   Round to 2 decimal places.
2. The null and alternative hypotheses for correlation are:
   H0:H0: ?rμρ == 0
   H1:H1: ?μρr ≠≠ 0
   The p-value is: (Round to four decimal places)
3. r2r2 = (Round to two decimal places)
4. The equation of the linear regression line is:   
   ˆyy^ = + xx   (Please show your answers to two decimal places)
   
5. Use the model to predict the weight of a woman who spends 46 minutes on the phone.
   Weight = (Please round your answer to the nearest whole number.)

3. The strength of the correlation motivates further examination.

a. Insert Scatter (X,Y) plot linked to the data on this s heet with Age on the horizontal (X) axis.

b. Add to your chart: the chart name, vertical axis label, and horizontal axis label

c. Complete the chart by adding Trendline and checking boxes: Display Equation on chart & Display R-squared value on chart

4. Read directly from the chart:

a. Intercept =

b. Slope =

c. R² =

Perform Data > Data Analysis > Regression

5. Highlight the Y-Intercept with yellow. Highlight the X variable in blue. Highlight the R Square in orange.

A professor's commute is normally distributed with a mean of 40 minutes and a standard deviation of 10 minutes. (a) What is the probability that the professor gets to work in 30 min or less? (Round your answer to three decimal places.) . (b) If the professor has a 9 A.M. class and leaves home at 8 A.M., how often is the professor late for class? (Round your answer to one decimal place.) - % of the time