For this project you will estimate parameters using confidence intervals. For each confidence interval, make sure you report  

• The confidence interval limits rounded as directed (Use StatCrunch to find all confidence interval limits. Use Stat>Proportion Stats to find confidence interval limits for proportions. Use Stat>T Stats to find confidence interval limits for means.) Refer to the resources on the course home page for more information about confidence intervals and how to find them using StatCrunch.
• An interpretation of the confidence interval.

Here is a template for reporting the answer for a sample problem.

Sample problem:

Find a 95% confidence interval for the proportion of all people who use their smart phones to watch movies.

Related Question: Does it appear 50% of all people use their smart phones for watching movies?

Interval Limits: 0.255 to 0.488

Interpretation: "I am 95% confident the interval from 0.255 to 0.488 contains the true value of the population proportion of people who use their smart phone to watch movies."

Related Question Answer: No, because 0.50 does not fall within the interval limits, therefore, 50% is not a plausible value for the proportion of all people who use their smarth phones to watch movies.

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Use the class data to find the confidence intervals described below. Type your answers in a Microsoft Word or rich text format document making sure you clearly show your answers for each problem and upload your document to submit your work. Use the Grapevine Online Statistics Data File in StatCrunch shared by user sgrapevine. Assume this data is representative of all online students. Round the interval limits as directed. Each problem is worth a total of 6 points (2 points for reporting the correct confidence interval limits, 2 points for providing a correct interpretation of the confidence interval, 1 point for answering the related question, and 1 point for explaining why or why not for each related question). You can also earn 1 point for rounding as directed for a total of 43 points. Note, this is not a team project. You must submit your own original work!

1) Find a 90% confidence interval for the mean hours of sleep online students typically get on a weekendnight. Round the interval limits to the nearest tenth.

Related Question: Is there strong evidence the average online student gets more than 7 hours of sleep on a weekend night? Why or why not?

2) Find a 95% confidence interval for the proportion of all online students that will call "heads" will given the option to call a coin toss. Round the interval limits to the nearest thousandth.

Related Question: Is it plausible that heads and tails are called equally when online students are asked to call a coin toss? Why or why not?

3) Find a 99% confidence interval for the mean foot length for all online students. Round the interval limits to the nearest tenth.

Related Question: Is there strong evidence the average foot length for all online students is different from 25 cm? Why or why not?

4) Find a 95% confidence interval for the mean number of letters in the last names of all online students. Round the interval limits to the nearest tenth.

Related Question: Is there strong evidence the average online student have a last name length of less than 10 letters? Why or why not?

5) Find a 90% confidence interval for the proportion of all online students who think dogs are the best pet. Round the interval limits to the nearest thousandth.

Related Question: Does it appear a majority (over 50%) of all online students think dogs make the best pet? Why or why not?

6) Find a 99% confidence interval for the proportion of all online students that pick the number 3 when asked to pick a number between 1 and 5. Round the interval limits to the nearest thousandth.

Related Question: Does it appear all online students select numbers randomly, that is, in this situation, is there strong evidence every number including the number 3 has the same 20% chance of being selected? Why or why not?

7) Find a 95% confidence interval for the proportion of all online students that have brown eyes. Round the interval limits to the nearest thousandth.

Related Question: Is it plausible 25% of all online students have brown eyes? Why or why not?

One Sample ttest:

The population average length of friendships with their closest female friend is known (µ= 69.61) but the population variance is unknown. As we learned from the text and the PPTs, we know these research situations require the use of the One Sample tTests. Here are the researchers’ hypotheses:

1. The researchers think that our group of 100 females will score differently than the population on the length of the friendship with their closest female friend.

Use the SPSS data to make conclusions.

Part 1—The Analyses

1. For this test you would run a One Sample tTests: one for “Length of the friendship with their closest female friend.” To do that, follow the instruction in Chapter 7. Here is an example and the output:

T-Test

Dependent (Paired) Sample ttest:

The researchers know that the rating of intimacy and emotional support will be similar. Because these ratings are made by the same people on the same friend, the scores are linked. As we learned from the text and the PPTs, we know these research situations require the use of the Paired Sample tTests.  Here are the researchers’ hypotheses:

1. The researchers think that our group of 100 females will rate intimacy and emotional support the same.  

Use the SPSS data to make conclusions.

Part 1—The Analyses

1. For this assignment, I ran a Paired Sample tTests for Intimacy and Emotional Support. Here is an example and the output:

T-Test

Part 2—The APA Write-Up Instructions

                  The American Psychological Association (APA) has standards for how statistical results should be presented.  While the actual word choice varies, there are several essential components that are common among all presentations of statistical results and interpretations.  Use the following instructions for ALL APA write-ups required for this course:

For each analysis, use the following steps to write a complete description of results in proper APA format.

1. State what hypothesis was tested.
2. State what test was used.
3. What decision did you make? Reject the null or retain (fail to reject) the null.
4. Were the groups significantly different from each other?
5. Report the means and standard deviations for each group.
6. Put numbers in APA format:
   1. General Format: symbol for the test (df)= obtained value, p> or < significance level
   2. Specific for tTests: t(99)=1.98, p<.05
7. Report Effect Size (if known)

Example:

         Adolescent males were expected to score significantly higher on a measure of aggression than were adolescent females. An independent ttest was used to test the hypothesis leading to the rejection of the null hypothesis. Adolescent males (M=14.67, SD=2.35) were significantly more aggressive than were adolescent females (M=9.65, SD=1.59), t(78)=3.93, p<.05.

Since we ran 2 analyses, you will need to do 2 write-ups for this assignment.

Analysis for One Sample ttest:

1. Answer the following questions
   1. What is the null hypothesis tested?
   2. Why is the One Sample tTest the correct test to use?
   3. Write up the results for the One Sample tTest you ran for Length of the friendship with their closest female friend below:  

Analysis for Dependent (Paired) ttest:

1. Answer the following questions
   1. What is the null hypothesis tested?
   2. Thinking about the original research hypothesis, does the researcher want to retain or reject the null in this case? Why?
   3. Write up the results for the Paired Sample tTest you ran Intimacy and Emotional Support below:  

Answer the following questions showing all work. Full credit will not be given to answers without work shown. If you use Minitab Express or StatKey include the appropriate output (copy + paste). If you do hand calculations show your work using the Word equation editor. Clearly identify your final answers. Output without explanation will not receive full credit and answers with no output or explanation will not receive full credit. Round all answers to 3 decimal places. If you have any questions, post them to the course discussion board.

1. A STAT 200 instructor wants to know if her students tend to score higher on the midterm exam than on the final exam. Data were collected from a representative sample of 46 students during the Fall 2017 semester. Data were paired by student. The mean difference, computed as midterm - final, was 1.3632 points with a standard deviation of 3.6362 points. [55 points]

A. In Minitab Express, conduct a paired means t test to determine if there is evidence that midterm exam scores are higher than final exam scores in the population of all STAT 200 students. Use the five-step hypothesis testing procedure and remember to include all relevant Minitab Express output. You should not need to do any hand calculations.

Step 1: Check assumptions and write hypotheses

Step 2: Calculate the test statistic

Step 3: Identify the p value

Step 4: Make a decision

Step 5: State a “real world” conclusion

B. In Minitab Express, conduct a single sample mean t test given a sample size of 46, sample mean of 1.3632, and sample standard deviation of 3.6362 to determine if there is evidence that the population mean is greater than 0. Use the five-step hypothesis testing procedure and remember to include all relevant Minitab Express output. You should not need to do any hand calculations.

Step 1: Check assumptions and write hypotheses

Step 2: Calculate the test statistic

Step 3: Identify the p value

Step 4: Make a decision

Step 5: State a “real world” conclusion

C. Explain why your test statistic and p-value were the same in parts B and C.

D. What minimum sample size would be necessary to construct a 95% confidence interval for the mean difference in midterm and final exam scores with a margin of error of 0.5 point? Show all work using the equation editor.

2. Download the BodyTemp.MTW file from Canvas. We will be comparing the body temperatures of men and women. [30 points]

A. Make a graph to compare the distributions of men and women’s body temperatures.

B. Use Minitab Express to determine if there is evidence that the mean body temperatures of men and women are different. The coding of gender is 0=man and 1=woman. Assume that the distribution of the body temperature data is normal. Use the five-step hypothesis testing procedure and remember to include all relevant Minitab Express output. You should not need to do any hand calculations.

Step 1: Check assumptions and write hypotheses

Step 2: Calculate the test statistic

Step 3: Identify the p value

Step 4: Make a decision

Step 5: State a “real world” conclusion

3. A random sample of Penn State World Campus undergraduate and graduate students were contacted and data concerning their gender identities were recorded. In a random sample of 40 undergraduate students, 21 identified as men and 19 identified as women. In a random sample of 40 graduate students, 17 identified as men and 23 identified as women. [15 points]

A. Is it appropriate to use the normal approximation method here to construct a confidence interval for the difference in population proportions? Show your work.

B. Construct a 95% confidence interval to compare the proportion of undergraduate students who identify as men to the proportion of graduate students who identify as men. If assumptions were met in part A, use the normal approximation method. Do not do any calculations by hand. Use Minitab Express and remember to copy+paste all relevant output and to clearly identify your final answer.

43 Inferential statistics, or hypothesis testing (i.e., t-tests, ANOVA, etc.), involves a comparison between which two elements?

• A. Research results from a sample and hypotheses about a population.
• B. Research results from a population and hypotheses about a sample.
• C. Research results from a sample and hypotheses about a sample.
• D. Research results from a population and hypotheses about a population.

4

A researcher wants to examine the effectiveness of different types of classroom instruction on students’ test performance. He designed two types of instruction for two different samples of students. Afterwards, both samples were given the same performance test and the test averages were compared. What statistical test should be used by the researcher to analyze the results of this study?

• A. t-test for repeated measures (dependent samples).
• B. One-factor ANOVA for independent samples.
• C. t-test for independent samples.
• D. Pearson Correlation

5 Dr. P assessed the relationship between the cost of different bottles of wine and the tastiness of the wine. Using a Pearson correlation test, the results of the study were r = -.32, n = 14, p >.05. Based on these results, which of the following is a correct conclusion.

• A. There is no significant correlation between the cost and tastiness of wine.
• B. There is a significant positive correlation between the cost and tastiness of wine.
• C. There is a significant negative correlation between the cost and tastiness of wine.
• D. Cannot be determined without additional information.

6 A researcher conducted a study and analyzes the data using an independent samples t-test. The results of the t-test were t(38) = 3.45, p <.05 and Cohen’s d = 0.80. Based on these results, which of the following would be a correct conclusion?

• A. There was a significant difference, with a small effect size.
• B. There was a significant difference, with a large effect size.
• C. There was no significant difference, with a small effect size.
• D. There was no significant difference, with a large effect size.

7

Which of the following is an accurate definition of Type I error in hypothesis testing?

• A. Failing to reject a true null hypothesis.
• B. Failing to reject a false null hypothesis.
• C. Rejecting a true null hypothesis.
• D. Rejecting a false null hypothesis.

8 A researcher used a sample of n=200 students to determine whether there are any significant differences in preference for five new exercise apps. Each participant was asked to upload and try each app and select his/her favorite. The number of students who favor each app was recorded. In order to test if there are any significant differences in preference for the new exercise apps, what statistical test should be used by the researcher?

• A. Two-factor ANOVA for independent samples.
• B. t-test for independent samples.
• C. Chi-Square test for Goodness of Fit.
• D. Chi-Square test for Independence.
• E. One-factor ANOVA for independent samples.
• F. Pearson correlation.

9 A research article reports the following: “The mean performance scores for Normal Sleep, Reduced Sleep, and No Sleep groups were 23.0, 16.6, and 11.6 respectively, F(2, 36) = 4.95, p < .05.” Based on this information, which conclusion is most accurate?

• A. There are some significant differences in performance among people who differ in amount of sleep.
• B. There are no significant differences in performance among people who differ in amount of sleep.
• C. People who sleep a reduced amount perform significantly worse than people who sleep a normal amount.
• D. People who get no sleep perform significantly worse than those who sleep a reduced amount.

50 Which of the following research examples could use a t-test for dependent samples (i.e., t- repeated-measures) for data analysis?

• A. A comparison of participants’ scores on attention focus test before and after attending a training session that was expected to improve the attention focus skill.
• B. A comparison of scores of participants in a memory study in which one group is assigned to learn the words in alphabetical order and another group is assigned to read the words in order of length of the word.
• C. A comparison of SAT scores of a group of 10 students who completed a special SAT preparation course compared to how people score on the SAT in general.
• D. None of the above.

Financial statement analysis is the process of analyzing a company's financial statements to make better economic decisions. The process for financial statement analysis includes specific techniques for evaluating risks, performance, and future prospects of an organization.

Directions

This Assessment is separated into four parts. In Part 1 of this Assessment, you will begin by researching and summarizing four of the benefits of financial analysis and indicating which is the most significant to you. In Part 2 of this Assessment, you will define the ratios listed. In Part 3 of this Assessment, you will classify, calculate, and explain the significance of Liquidity, Profitability, and Market Value financial ratios, and provide a year to year comparison of assessed financial trends. In Part 4 of this Assessment, you will compose an analytical essay in a minimum of 350 words reporting the one financial measurement trend you find to be most significant and actions necessary to improve results.

Use this Word template provided to complete your Assessment.

Questions

Part 1.

Locate and read the following article located from the Library:

Faello, J. (2015). Understanding the limitations of financial ratios. Academy of Accounting & Financial Studies Journal, 19(3), 75–85.

Refer to pages 75 and 76 of Faello’s (2015) work. In four separate paragraphs (one for each question), summarize four of the benefits of financial analysis mentioned in the journal article. In one paragraph, describe which one of the four benefits you consider to be most significant.

Review the following financial data, and then answer the questions below.

Part 2.

Define the ten financial ratios below.

Financial Ratios:

Financial Ratios:

Current Ratio is equal to Current assets / Current Liability

2014: 61,185,000 / 69,345,000 = 96.95%

Quick Ratio) (Cash and cash equivalents + Current receivable) / Current Liability

2014: 7,281,000 + 6,677,000 / 69,345,000 = 20.13%

2015: 9,135,000 = 6,778,000 / 65,272,000 = 24.38%

Accounts Receivable Turns) Net Credit Sales / Average Accounts Receivable

485651000 / 6,727,500 = 72.19

Accounts Payable Turns

Return on Equity) Net Income / Shareholder’s equity

16,303,000/81,394,000 = 20.03%

Return on Assets

Operating Profit Margin

Net Profit (after tax) Margin

Earnings per Share

Price to Earnings

Question 1: Activity-Based Costing (ABC)

Doris Ltd manufactures two models of medical devices for blood testing as follows: N80 and N90. The company is currently using the traditional (absorption) costing method and overhead is applied to the products based on the number of units. Doris Ltd has decided to adopt activity-based costing (ABC) in the coming year. The company data concerning the current period’soperations appear below:

From the company’s management information system, you extract the following information on activities, costs and activity drivers for the current period:

5

Question 1 continued:

Required:

1. Compute the predetermined overhead rate under the current absorption costing method and determine the full unit product cost of each product. Round the unit costs and all other costs to 4 decimal places.

2. Determine the full unit product cost of each product using the activity-based costing approach. Round the unit costs and all other costs to 4 decimal places.

   (Total marks for the question 32 marks)

Enter your answers for the following sub-questions in Part A on Canvas. Enter all your answers as numbers without the $ sign. Ensure you show the working for all sub-parts in your Part B Word document.

Question 1

For a party tonight you want to make your special hamburger recipe. You have only the following ingredients:

10 meat patties

24 buns

18 cheese slices

If you need to make as many complete hamburgers as possible and each hamburger must follow your special hamburger recipe contain 1 meat patty, 2 bun halves, and 2 cheese slice, then what is the maximum number of complete hamburgers can you make given the reagents, I mean ingredients, that you have on hand?

You can make a maximum of 24 hamburgers.
You can make a maximum of 9 hamburgers.
You can make a maximum of 10 hamburgers.
You can make a maximum of 18 hamburgers.
You can make a maximum of 12 hamburgers.  

QUESTION 2

Write down logical dimensional analysis steps, starting with the given number of each ingredient and ending up with numbers of hamburgers; match the number of food reagents available with the number of hamburgers just that reagent could make if that ingredient is all that controlled your hamburger production that day.

10 meat patties

33 tomato slices

18 cheese slices

24 buns halves

A. 9 hamburgers

B.12 hamburgers

C. Wait a minute! Tomatoes aren't in this receipe.

D.10 hamburgers

QUESTION 3

Given the food supplies for the party (the reagents) in problem #1 above. Which "reagent" would make the largest number of hamburgers if the "yield" were only based on that reagent? Note this is NOT the limiting reagent. This is the fantasy scenario. If you follow the recipe, it is impossible to make this number of hamburgers because one of the other reagents will be used up.

The buns could make 24 complete hamburgers.
The patties could make 20 complete hamburgers.
The cheese could make 18 complete hamburgers.
The cheese could make 9 complete hamburgers.
The buns could make 12 complete hamburgers.
The patties could make 10 complete hamburgers.

QUESTION 4

Given the food supplies for the party (the reagents) in problem #1 above. Which "reagent" would make the fewest number of hamburgers if the "yield" were only based on that reagent and you followed the recipe? The reagent you choose is called the limiting reagent because it makes the least amount of the product. It makes the least amount of product because it's completely consumed.

The buns would make 12 complete hamburgers.
The patties would make 10 complete hamburgers.
The cheese would make 18 complete hamburgers.
The cheese would make 9 complete hamburgers.

QUESTION 5

Given the food reagents in Problem #1, after the maximum number of hamburgers are made, what number of each excess reagent will be left over?

Here's how to do this: start with the number of the limiting reagent and calculate the number of each of the other reagents CONSUMED when the limiting reagent is completely used up. Subtract those from the number your started with and that's what's left over for that reagent. All ingredients will be left over EXCEPT for the limiting reagent.

1 meat patty, 0 cheese slices, and 6 bun halves
9 meat patties, 9 cheese slices, and 2 bun halves
9 patties, 2 cheese slices, and 12 bun halves
0 patties, 5 cheese slices, and 10 bun halves

QUESTION 6

This limiting reagent hamburger quiz is analogous to the chemical limiting reagent problems we've been studying in many ways. Match the following correctly:

The word must in the first problem describing that each hamburger must be composed of a certain recipe corresponds to:

Determining the food item that will make the fewest number of hamburgers corresponds to:

Determining which and how much of each food items will be left over after the hamburgers are made corresponds to:

Determining which food item makes the most hamburgers corresponds to:

A. the process of determining the limiting reagent in chemical analysis.
B. nothing useful in chemical analysis.
C. the stoichiometric relationship between reactants and products.
D. the process of determining the excess reagents in chemical analysis.

Getting It Right: The Success of Continental Airlines

Continental Airlines was doing something that seemed like a horrible mistake. All other airlines at the time were following a simple rule: They would only offer a flight if, on average, 65 percent of the seats could be filled with passengers, since only then could the flight break even. Continental, however, was flying jets at just 50 percent of capacity and was actually expanding flights on many routes. When word of Continental’s policy leaked out, its stockholders were angry, and managers at competing airlines smiled knowingly, waiting for Continental to fail. Yet Continental’s profits – already higher than the industry average – continued to grow. What was going on?

            There was, indeed, a serious mistake being made – but by the other airlines, not Continental. This mistake should by now be familiar to you [students of managerial economics]: using average cost instead of marginal cost to make decisions. The “65 percent capacity” rule used throughout the industry was derived more or less as follows: The total costs of the airline for the year (TC), was divided by the numbers of flights during the year (Q) to obtain the average cost of a flight (TC/Q=ATC). For the typical flight, this came to about $4,000. Since a jet had to be 65 percent full in order to earn ticket sales of $4,000, the industry regarded any flight that repeatedly took off with less than 65 percent as a money loser and canceled it.

            As usual, there are two problems with using ATC in this way. First, an airline’s average cost per flight includes many costs that are fixed and are therefore irrelevant to the decision to add or subtract a flight. These include the cost of running the reservations system, paying interest on the firm’s debt, and fixed fees for landing rights at airports – none of which would change if the firm added or subtracted a flight. Also, average cost ordinarily changes as output changes, so it is wrong to assume it is constant in decisions about changing output.

            Continental’s management, led by its vice-president of operations, had decided to try the marginal approach to profit. Whenever a new flight was being considered, every department within the company was asked to determine the additional cost they would have to bear. Of course, the only additional costs were for additional variable inputs, such as additional flight attendants, ground crew personnel, in-flight meals, and jet fuel. These additional costs came to only about $2,000 per flight. Thus, the marginal cost of an additional flight – $2,000 – was significantly less than the marginal revenue of a flight filled to 65 percent of capacity – $4,000. The marginal approach to profits tell us that when MR>MC, output should be increased, which is just what Continental was doing. Indeed, Continental correctly drew the conclusion that the marginal revenue of a flight filled at even 50 percent of capacity – $3,000 – was still greater than its marginal cost, and so offering the flight would increase profit. This is why Continental was expanding routes when it could fill only 50 percent of its seats.

            In the early 1960’s, Continental was able to outperform its competitors by using a secret – the marginal approach to profits. Today, of course, the secret is out, and all airlines use the marginal approach when deciding which flights to offer.

After reviewing chapter 8, I am sure you will come to appreciate the significance of the marginal analysis for optimal decisions first introduced in Chapter 1. In Chapter 8, the approach boils down to what we generally call the optimal output decision rule that states: To maximize profit or minimize losses in the short run, a firm should produce at the level where marginal revenue (MR) is equal to marginal cost (MC).  Although this principle was long established by economists, the business community, however, was initially slow to understand and use it in their decision making processes. Those who understood the method and applied it earlier than their competitors were able to generate substantial profits while their competitors struggled to survive. Please find attached a summary of an article that demonstrates these events and along the way demonstrates how useful and powerful the method is.

After reviewing the summary, please post your thoughts and reactions. In particular, anything that you found particularly surprising or interesting and some other mistakes managers make, in light of what you have learned in this module. no graph needed.

Project 2
This assignment follows the standard form for a project submission. You need to include an introduction, primary discussion, and summary. Include graphs, tables, and images, as necessary, to improve the clarity of your discussion. Your project needs to be both correct and well written. Communication remains a critical component of our modern, technological society. A few notes about format: you MUST use MS Word for your project and use Equation Editor for all mathematical symbols, e.g. z(t) = sin(t) +   1/ln(t)
If you have any questions about the requirements for this project, ask before you submit.

This project addresses modeling with Ordinary Differential Equations and solutions to those equations. You will solve a problem analytically and program an Improved Euler’s method numerical solver.

Projects provide you with an opportunity to improve your Mathematical skills as well as your communication. For this project you will need to correctly solve the problems and effectively
communicate your ideas and solutions. This assignment will be evaluated across the areas of
Validity, Readability, and Fluency.
Validity – Validity corresponds to the validity of your arguments. It addresses the extent to which your method is appropriate, your calculations are correct, and your analysis is accurate.
Readability – If your written work is not readable it cannot be assessed. Since the ability to communicate Mathematics is a focal point for this class, special attention will be paid to the
readability of your work.
Fluency – Mathematics is a concise and precise language, and we wish to enhance your fluency.
Therefore, part of every assessment will focus on your ability to incorporate correct, established notation and terminology into your written work

Evaluation criteria
Validity              quality methods, correct solutions, proper conclusions, complete reasoning
Readability       organization, presentation, format, clarity, effectiveness
Fluency              proper notation, proper terminology, appropriate definitions, conciseness

Project 2: A bead sliding along a rod

A bead is constrained to slide along a rod of length L. The rod is rotating in a vertical plane with a constant angular speed, W, about a pivot in the middle of the rod. The pivot allows the
bead to freely slide along the rod, i.e. the pivot does not impede the movement of the bead. Let r(t) denote the distance of the bead away from the pivot where r(t) can be positive
or negative.

Equation of Motion

Applying Newton’s second law provides a balance of forces due to gravity, friction, centripetal acceleration, and linear acceleration. The equation resulting from these forces is

M d²r/dt² + B dr/dt – mw²r = -mg sin(wt) where m is the mass of the bead, B is the coefficient of viscous damping, w is the constant speed of angular rotation, g = 9.81 m/s² is the acceleration due to gravity, and r is the distance between the pivot and the bead. The rod is initially horizontal, and the initial conditions for the bead are r(0) = r₀ and r’(0) = v₀.

Problem 2

Consider the frictionless rod, i.e. B = 0. The equation of motion becomes m d²r/dt² - mw²r = -mg sin (wt) with g = 9.81 m/s² and a constant angular speed W. The rod is initially horizontal, and the initial conditions for the bead are r(0) = r₀ and r’(0) = v₀. You will need to write an Improved Euler Method system solver to find r(t) and v(t)

A) Numerically solve for r(t) when W = 2, r₀ = 0, and v₀ = 2.40, 2.45, 2.50. Solve in the time interval t E [0,5] . Use step sizes h = 1/32, 1/128, 1/512 and compare your results. Also, compare your best numerical answers with your analytic answers from Problem 1 part E).
B) Numerically solve for r(t) when W = 2, r₀ = 0, and v₀ is selected to give simple harmonic motion, i.e. Problem 1 part B. Use small step sizes, e.g. h =  1/512, 1/2048, 1/8192, etc. Solve for the longest time interval that provides reasonable values for r(t) . Compare your results to the analytic solution that gives simple harmonic motion. What does this demonstrate about numerical solutions?