Below is a list of the profit or (loss) of 40 companies.

- Calculate the population parameters for all 40 companies profit or loss

- Select a sample of 10

- Calculate the point estimators of the sample

- Compare the results (Point estimators with Population Parameters)

$ In Millions

$9,862             $19,710          

$44,940           $48,351

$10,558           $5,070

$6,662             $3,033

$29,450           ($3,864)

7,602               $364.5

$9,195             $1,288

$2,679             $30,101           

$1,907             ($5786)

$4,078             $24,441

$2,463             $12,662

$8,630             $18,232

$4,517.4          $22,183

$8,197             $5,106

$3,842.8          $21,204

$4,065             $22,714

$2,997             $9,609

$246.5             $4,286

$3,577             $2,736

$2,421.9          $1,982

Question 3 This question is testing your understanding of some important concepts about hypothesis testing and confidence intervals. For each part below, answer the question (1 mark) and then succinctly explain your reasoning (1 mark). (a) We are performing a one-sample t test at the 5% level of significance where the hypotheses are 0 1 H VH :5 :5 µ µ = ≠ . The number of observations is 15. State the critical value? (b) We are performing a hypothesis test and we conclude that we reject H0 at the 5% level of significance. Will we reject the same H0 (with the same H1 ) at the 10% level of significance? (c) Suppose we are performing a hypothesis test and we conclude that we cannot reject H0 at the 5% level of significance. Can we reject the same H0 (with the same H1 ) at the 10% level of significance? (d) Suppose we are performing a two-sample proportion test at the 5% level of significance where the hypotheses are 01 2 11 2 H p p VH p p : 0: 0 −= −≠ . The calculated p-value is 0.00268. Do we reject H0 ? (e) Based on the data, we obtain (1.85, 1.95) as the 95% confidence interval for the true mean. Can we reject 0 H : 0 µ = against 1 H : 0 µ ≠ at the 5% level of significance?

Problem 1.

(a) The columns of response and factors can be defined in R as follows, use these codes to solve the problem.

y<-c(2, 3, 10, 12, 8, 4, 11, 8)##response : scores

a<-c("Heart","Heart", "Soul", "Soul","Heart","Heart", "Soul", "Soul")##factor A

b<-c("D", "D", "D", "D", "R", "R", "R", "R")##factor B (group variable)

(b) Find the overall mean, row means, column means, each cell mean for the table given in problem 1.

A data set includes 103 body temperatures of healthy adult humans having a mean of 98.5°F and a standard deviation of 0.61°F. Construct a 99​% confidence interval estimate of the mean body temperature of all healthy humans. What does the sample suggest about the use of 98.6°F as the mean body​ temperature?

What is the confidence interval estimate of the population mean μ​?

°F < μ < °F

​(Round to three decimal places as​ needed.)

When applying statistical tests involving comparing two means or a sample to a population mean, there are many organizational applications. For example, Human Resources may want to track entrance exam scores of their new hires. This would be an example of two mean comparison. In terms of the recent election, Gallup may take a sample and compare to a population of candidate votes (sample mean compared to a population mean).

Think of an example in your organization of either one of these tests and discuss its application as well as the risk of type 1 or 2 errors.

2. What is the sample size, n, for a 95% confidence interval on the mean, if we know that the process’ standard error is 3.2 units, and we want to allow at most 1.0 units for our error?

3. Let’s say that you just randomly pulled 32 widgets from your production line and you determined that you need a sample size of 46 widgets, However, you get delayed in being able to pull another bunch of widgets from the line until the start of the next day. How many widgets should you now pull for your analysis?

4. What is the sample size, n, for a 98% confidence interval on the mean, if we know that the process’ standard error is 3.2 units, and we want to allow at most 0.5 units for our error?

5. What is the sample size, n, for a 95% confidence interval on the mean, if we know that the process’ standard error is 3.2 units, and we want to allow at most 0.5 units for our error?

According to published reports, practice under fatigued conditions distorts mechanisms that govern performance. An experiment was conducted using
15 college males, who were trained to make a continuous horizontal right-to-left arm movement from a microswitch to a barrier, knocking over the barrier coincident with the arrival of a clock sweephand to the 6 o’clock position. Theabsolute value of the difference between the times, in milliseconds, that it took to knock over the barrier and the time for the sweephand to reach the 6 o’clockposition (500 msec) was recorded. Each participant performed the task five times under prefatigue and postfatigue conditions, and the sums of the absolute differences for the five performances were recorded. The data can be found in the folder of this question.

1. a) (0.5 point) Read the data into R using read.csv function. Note: Show your codes but not the result/output.

2. b) (0.5 point) An increase in the mean absolute time difference when the task is performed under postfatigue conditions would support the claim that practice under fatigued conditions distorts mechanisms that govern performance. Assuming the populations to be normally distributed, write the two hypothesis of interest to test this claim.

c) (1 point) Use a suitable test in R to test your hypothesis in (b). Show your codes, output and use α = 0.05.

d) (1 point) Interpret your finding in (c).

data:

Prefatigue,Postfatigue
159,92
93,60
66,216
99,227
34,224
90,92
149,93
59,178
143,135
118,117
75,154
67,220
110,144
58,165
86,101

A machine that puts corn flakes into boxes is adjusted to put an average of 15.1 ounces into each box, with standard deviation of 0.23 ounce. If a random sample of 15 boxes gave a sample standard deviation of 0.35 ounce, do these data support the claim that the variance has increased and the machine needs to be brought back into adjustment? (Use a 0.01 level of significance.)

(i) Give the value of the level of significance.


State the null and alternate hypotheses.

H₀: σ² < 0.0529; H₁: σ² = 0.0529

H₀: σ² = 0.0529; H₁: σ² ≠ 0.0529

    H₀: σ² = 0.0529; H₁: σ² < 0.0529

H₀: σ² = 0.0529; H₁: σ² > 0.0529

(ii) Find the sample test statistic. (Round your answer to two decimal places.)


(iii) Find or estimate the P-value of the sample test statistic.

P-value > 0.1000

.050 < P-value < 0.100

    0.025 < P-value < 0.0500

.010 < P-value < 0.0250

.005 < P-value < 0.010

P-value < 0.005

(iv) Conclude the test.

Since the P-value ≥ α, we fail to reject the null hypothesis.

Since the P-value < α, we reject the null hypothesis.

   Since the P-value < α, we fail to reject the null hypothesis

.Since the P-value ≥ α, we reject the null hypothesis.

(v) Interpret the conclusion in the context of the application.

At the 1% level of significance, there is sufficient evidence to conclude that the variance has increased and the machine needs to be adjusted.

At the 1% level of significance, there is insufficient evidence to conclude that the variance has increased and the machine needs to be adjusted.    

Results from the National Health Interview Survey show that among the U.S. adult population, 45.9% do not meet physical activity guidelines, 3.5% meet only strength activity, 29.0% meet only aerobic activity, and 21.6% meet both strength and aerobic activity. We sampled 4475 adults from Ohio and the results were as follows: 50.0% do not meet physical activity guidelines, 5.0% meet only strength activity, 35.0% meet only aerobic activity, and 10.0% meet both strength and aerobic activity. Conduct an appropriate hypothesis test to determine if the distribution in physical activity among Ohioans is similar to the U.S. population. Interpret your results.

PLEASE USE RULES OF ADDITION OF PROBABILITY

/MDPART 1: Create a context: Introduce a real-world business situation or an imagined business scenario that may benefit from the theories and methods presented in the prior two weeks (Weeks 3 and 4 - see the objective stated in bold above). While creating the context, ask yourself: What business use cases may benefit from the quantitative methods offered last week and this week? Pick one. You can use Google or other search tools, or your own workplace experience, to create the context. Once you decide on the context, describe it clearly in your PPT presentation. Be sure to deliver all relevant information for the audience (in this case your instructor) to help him/her understand the context fully.

Part 2: Show with examples how the quantitative methodology introduced in the prior two weeks relate to the business use case you introduced. (e.g., does the math/methodology help to resolve a business problem?) Be specific in your description of how you can use the quantitative information in connection with the business case (your context) you described. Give examples. Use technical terminology when necessary.

Part 3: Reflect on your learning by answering these questions: What changes have you observed in your own learning or knowledge of math/quantitative methods as a result of the topics introduced in Weeks 3 and 4 of this course? What did you find most valuable or useful for your MBA education and/or your current/future career. (As you articulate your thoughts for

Part 3, be original: Do not repeat the business context or situation you described in Parts 1 and 2. Think beyond that context

To use Excel INV functions, such as XXXX.INV(probability,....) to generate random variates, replace probability by:

		the mean of the distribution
		0.05, the level of significance
		the uniform distribution formula
		RAND()

For the following situation identify the sampling method used.

The Florida Fish and Wildlife Conservation Commission manatee mortality database website has information on cause of death and size for all manatees killed off Florida waters.We decide to divide the cause of death into watercraft, natural causes, stress, and other. Within each of these groups, a random sample of 25 manatees are selected.

	A.	systematic sampling
	B.	stratified sampling
	C.	simple random sampling
	D.	cluster sampling

Use the data for two Sydney suburbs to answer questions 1-5:            

1     Using Excel or other appropriate software, produce two separate histograms for the median rental prices of the two selected suburbs, respectively.    

       Use an appropriate number of bins for your histogram and remember to label the axes. Describe and compare the two histograms, including the central location, dispersion and skewness. [1 mark]     

            

2     Compute the sample means and sample standard deviations of the median rental prices of the two selected suburbs, respectively. No need to show computational steps. [1 mark]   

Artarmon	Chatswood
570	730
660	680
660	920
595	1100
500	880
610	840
600	750
600	540
600	490
590	630
690	600
625	600
610	1200
600	900
595	800
550	730
680	700
500	675
580	660
650	580
530	820
550	700
490	680
665	610
680	1200
580	900
680	880
670	870
585	850
660	800
500	800
595	790
490	780
530	780
565	750
570	750
580	750
595	720
600	720
600	700
610	680
625	660
660	650
680	650
690	650
580	610
585	600
650	595
665	580
590	580

I am having trouble with all three parts.

A shelf contains 3 novels, 2 books of poetry, and 1 dictionary. We select 2 books at random in turn without replacement. Define events A and N by: A = “the dictionary is selected”, N = “at least one novel is selected”. Show your work to find each of the following:

(a) P(A' )

(b) P(N)

(c) P(A ∩ N).

An outbreak of Salmonella-related illness was attributed to ice cream produced at a certain factory. Scientists are interested to know whether the mean level of Salmonella in the ice cream is greater than 0.2 MPN/g. A random sample of 20 batches of ice cream was selected and the level of Salmonella measured. The levels (in MPN/g) were:

0.593, 0.142, 0.329, 0.691, 0.231, 0.793, 0.519, 0.392, 0.418, 0.219 0.684, 0.253, 0.439, 0.782, 0.333, 0.894, 0.623, 0.445, 0.521, 0.544

a) (0.5 point) Read the data in R using a vector. Show your codes only but not the output.

b) (0.5 point) State the two hypotheses of interest.

c) (1 point) Is there evidence that the mean level of Salmonella in the icecream is greater than 0.2 MPN/g? Assume a Normal distribution and use α =0.05. Show your codes and result/output from R.

d) (1 point) Interpret your finding in (c).