Assignment 6 consists of a theory component (6A, worth 80%) and a computer component (6B, worth 20%).
The grades allocated are summarized below.
|Assignment 6A. Theory component||60 marks||80%|
|Assignment 6B. Computer component||15 marks||20%|
|Total marks||75 marks||100%|
You must submit Assignment 6A and Assignment 6B together as PDFs, through the appropriate drop box on the course home page. Submit:
- One PDF solution file (file) entitled Assignment6A containing all your answers to Assignment 6A, presented in the proper order. Your name and student ID number must be at the top of the first page of your solution file.
- One PDF solution file (file) entitled Assignment6B containing all your answers to Assignment 6B, presented in the proper order.
We suggest that you print the Assignment 6 questions, so that you can conveniently review the questions with their solutions when you prepare for your exams.
Assignment 6A. Theory Component
Show your work for this component. Do not round off your middle work. Keep your final answer to four decimals, where relevant.
Problem 1. Gestation (26 marks)
The data below are the gestation periods, in months, of randomly selected animals and their corresponding lifespans, in years.
Figure 1. Gestation data.
- If you want to predict the lifespan of an animal with a certain length of gestation period, which variable will be the response variable and which one will be the explanatory variable? [2 marks]
- Compute the correlation coefficient between the gestation period and the lifespan. Interpret the value. [4 marks]
- At α = 0.05, can you conclude that there is a significant correlation between the two variables? [6 marks] (Hint: Show the key steps of test of Hypothesis.)
- Compute the coefficient of determination r2. Interpret the value. [3 marks]
- Is it gestation period alone that determines lifespan? Are there other factors that need to be considered? Justify your answer. [1 mark]
- Use the method of least squares, compute the linear regression equation that will help you predict the lifespan from the gestation period. [5 marks]
- Interpret the slope of the regression equation with reference to context. [2 marks]
- Predict the lifespan of a moose, which has an 8-month gestation period. Also, compute the residual for this lifespan. [3 marks]
Problem 2. Mileage (10 marks)
A manager wishes to determine the relationship between the distance (in hundreds of miles) the manager’s sales representatives travel per month and the amount of sales (in thousands of dollars) per month.
Figure 1. Mileage data.
- Find the equation of the regression line for the given data. [5 marks]
- Find the standard error of estimate. Interpret the value. [5 marks]
Problem 3. Walkers (10 marks)
A medical researcher is interested in determining if there is a relationship between blood pressure and regular walk for adults over 50. A random sample of 240 adults over 50 is selected and the results are given below. Test whether the blood pressure is independent of regular walk. (Use α = 0.05.)
Figure 1. Blood pressure data.
- Identify the claim and state the null and alternative hypothesis. [2 marks]
- Determine the critical value and the rejection region. [2 marks]
- Calculate the test statistic. [4 marks]
- Decide whether to reject or fail to reject the null hypothesis. Interpret the decision in the context of the problem. [2 marks]
Problem 4. ANOVA (4 marks)
Fill in the missing entries in the following partially completed one-way ANOVA table. [4 marks]
|Variation||Sum of Squares||Degrees of Freedom||Mean Squares||F Statistic|
Problem 5. Square footage (10 marks)
A realtor wishes to compare the square footage of houses of similar prices in 4 different cities. He took a sample of 5 houses from City 1, 4 houses from City 2, 6 houses from City 3, and 7 houses from City 4. Sum of Squares Total is 71.06 and Mean Squares Between is 8.75. Can the realtor conclude that the mean square footage is the same in all four cities? Assume that the level of significance is 0.01. [6 marks]
(Hint: Set up an ANOVA table with the given information and then complete the ANOVA table for the missing information. Also include the key steps of a test of hypothesis question). [4 marks]
Assignment 6B. Computer Component
Note: Where relevant, do NOT round off the results you get from StatCrunch.
Problem 1. Funland: Questionnaire
In Assignment 4B, you worked with Funland’s questionnaire data. This problem asks you to continue this work. The information is repeated here for your convenience.
Funland, an indoor amusement park located in a large mall, offers midway rides, games, fast foods, and beverages. The owners of Funland distributed the survey questionnaire to its regular customers. Twenty-five regular customers responded to this survey, and their responses are stored in the StatCrunch file Funland in the Math 216 group folder on StatCrunch.
|For questions 1-4, please select the responses that apply to you.|
For questions 5-8, please write the number that applies.
|1. Please indicate your marital status.||Marital|
|2. Please indicate your gender.||Gender||Code|
|3. Do you frequently bring children to Funland?||Child||Code|
|4. Do you purchase a monthly pass?||Pass||Code|
|5. On a typical visit, please estimate the amount that you spend at Funland.||Spend||$ _______|
|6. How many times per month do you typically visit Funland?||Visits||_______|
|7. Please indicate your monthly family income before taxes.||Income||$ _______|
|8. Please indicate your age.||Age||_______|
Open the StatCrunch data file Funland and answer the following.
- Use StatCrunch to create a contingency table for the survey responses in the StatCrunch Funland data file, with Recode(Gender) displayed as the row variable, and Recode(Pass) displayed as the column variable. Copy and paste the Contingency Table created from StatCrunch to the Word file Assignment6B under the subheading Problem 1a. [1 mark]
- Use the contents of the Chi-Square Test Table to conduct a test of hypothesis, at a 5% level of significance” to determine whether the variables Recode(Gender) and Recode(Pass) are independent or related. Use the four-step P-value approach. Type the Chi-Square Test Statistic and related P-value in Step 2 of the test in the Word file Assignment6B under the subheading Problem 1b. [1 mark]
- Use the row percents displayed in the second column (Yes column) of the contingency table to describe the relationship between the variables Recode(Gender) and Recode(Pass). Type your answer in the Word file Assignment6B under the subheading Problem 1c. [0.5 marks]
- Comment on the Expected Frequency assumption underlying the Chi-square test used in independence test used in Problem 1b. Type your comments in the Word file Assignment6B under the subheading Problem 1d. [0.5 marks]
Problem 2. Engineering student: Grade Point Average
A university researcher wants to determine whether there is a significant linear equation relating a university engineering student’s first year grade point average (variable Yr1GPA) with that student’s Grade 12 high school math grade (variable G12Grade). A random sample of 100 pairs of Year 1 GPAS and Grade 12 math grades (for 100 different first year engineering students) is shown in the StatCrunch data file EngineerStudent, available in the StatCrunch Math 216 groups folder.
Open the file EngineerStudent and use StatCrunch to do the following:
- Find the equation of the linear regression line with G12Grade being the independent variable. Copy and paste the first screen of the Simple Linear Regression Results window from StatCrunch to the Word file Assignment6B under the subheading Problem 2a. Type the linear regression equation under the pasted Simple Linear Regression Results window. [1 mark]
- Plot the regression line along with the scatterplot. Copy and paste the graph in the second screen of the Simple Linear Regression Results window to the Word file Assignment6B under the subheading
Problem 2b. [1 mark]
- What does the slope of the regression line suggest about the relationship between the two variables? Type your answer in the Word file Assignment6B under the subheading Problem 2c. [0.5 marks]
- Based on the Simple Linear Regression Results window pasted from 2a, Type the Coefficient of Determination, r-squared, in the Word file Assignment6B under the subheading Problem 2d. [0.5 marks]
- Interpret the Coefficient of Determination number computed in 2d. Type your answer in the Word file Assignment6B under the subheading Problem 2e. [0.5 marks]
- Test to see if the slope of the regression line significantly exceeds zero. Use the four-step P-value approach. Based on the Simple Linear Regression Results window from 2a, type the P-value in the Word file Assignment6B under the subheading Problem 2f, under Step 2 of the hypothesis test. Assume a level of significance of 5%. [1 mark]
- Construct a 95% prediction interval for the first-year GPA for a university engineering student who got a 79%grade in her Grade 12 math course. Based on the Simple Linear Regression Results window pasted from 2a, Type the prediction interval in the Word file Assignment6B under the subheading Problem 2g. [1 mark]
Problem 3. Engineering student: Working with data
Open the StatCrunch file EngineerStudent file, available in the StatCrunch Math 216 group folder.
- Use StatCrunch to create a Scatterplot with G12Grade on the X axis and Yr1GPA on the Y axis. Copy and paste the Scatterplot graph in to the Word file Assignment6B under the subheading Problem 3a.
- Use StatCrunch to compute the correlation coefficient, r, between G12Grade and Yr1GPA. Copy and paste the Correlation Coefficient output window to the Word file Assignment6B under the subheading Problem 3b. [1 mark].
- Interpret the Correlation Coefficient number computed in 3b, in the context of the two variables, G12Grade and Yr1GPA. Type your answer in the Word file Assignment6B under the subheading
Problem 3c. [0.5 marks]
- At a 5% level of significance, use StatCrunch to conduct the t-test to see if the population correlation coefficient between G12Grade and Yr1GPA, is significantly different from zero. Use the four-step
P-value approach. Copy and paste the Hypothesis Test Results window from StatCrunch to the Word file Assignment6B under the subheading Problem 3d under Step 2 of the test. [1 mark]
Problem 4. Hospital stays
A provincial health board has hired you to conduct a study comparing the hospital stay of patients admitted to one of three hospitals in a large Canadian city: Hospitals A, B, and C. You have randomly selected three samples of patients (one sample from each hospital) and recorded the number of days each patient stayed at the hospital after being admitted. This data is shown in the StatCrunch data file Hospital, available in the StatCrunch Math 216 groups folder.
Open the file Hospital and use StatCrunch to do the following:
- Conduct an ANOVA test of hypothesis to see if the mean number of days spent in the hospital per patient differs between the three hospitals, at a 1% level of significance. Use the four-step P‑value approach. Copy the ANOVA Results Table under Step 2 of the ANOVA test in the Word file Assignment6B under the subheading Problem 4a. [1 mark]
- Test the assumption of equal variances at a 1% level of significance. Use the four-step P-value Approach. Copy and paste the table displaying the Levene’s Test for Homogeneity of Variance to the Word file Assignment6B under the subheading Problem 4b, under Step 2 of the test. [1 mark]
- Use the Shapiro-Wilk Test for Normality to determine if the 3 samples in the dataset Hospital all appear to come from normal populations. Assume a 1% level of significance for this test. Use the four-step P-value Approach. Copy and paste the Shapiro-Wilk goodness-of-fit results table to the Word file Assignment6B under the subheading Problem 4c, under Step 2 of the test. [1 mark]
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