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MDM4U1 Assignment 3- Two Variable StatisticsName:______________________ KU: /13 APP: /19 COMM: /4 TI: /KnowledgeMultiple Choice (6 Marks) Identify the choice that best completes the statement or answers the question.a 1. Which set of data would probably show a strong negative linear correlation? a. ...

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M. Dadar Solution

Assignment 3- Two Variable Statistics

MDM4U1

Name:______________________ KU: /13 APP:

/19

COMM: /4 TI:

/12

Knowledge Multiple Choice (6 Marks) Identify the choice that best completes the statement or answers the question. __a__ 1. Which set of data would probably show a strong negative linear correlation? a. resale values of computers and their ages b. heights volleyball players can jump and the strength of their leg muscles c. numbers of people at a water park and the air temperature d. scores on a mathematics test and the number of hours spent studying for it _ b___ 2. _______ is a point in a set of data that is significantly far from the majority of the other data. a. Residual b. Outlier c. Correlation d. Median __d__ 3. Using a linear-regression equation to predict values inside the range of the data is an example of a. extrapolation c. least-squares fit b. residuals d. interpolation __c__ 4. Two variables have a coefficient of determination of 0.25. The correlation coefficient could be: a. 0.25 b. 0.0625 c. – 0.5 d. -0.25 __b__ 5. A data set with coefficient of determination of 0.6 has? a. Moderate positive correlation b. Strong positive correlation. c. Weak positive correlation d. We cannot tell the strength of the correlation __d__ 6. Which of the following statements is true? a. Residuals are outliers that may be eliminated from the calculations. b. Residuals are positive for points below the line of best fit. c. For a line of best fit, the sum of the squares of the residuals is zero. d. For a line of best fit, the sum of the residuals is zero.

7. Select the dependent variable that seems most appropriate for each of the following pairs of variables. (1 mark each) a) The age of manufacturing equipment and the number of rejects produced by the equipment. Number of rejects b) Interest rates and the level of foreign investment in Canada. Level of foreign investment c) Sales volume and the number of marketing representatives. Sales volume

M. Dadar Solution

8. State whether the following scatter diagrams look like linear relationships or curvilinear relationships. In the case of a linear relationship, indicate whether the slope of the line is

Curvilinear

Linear – Positive

Curvilinear

Application

9. Carla has kept track of the hours she spent studying and her marks on examinations. The following table shows the results: [11 marks] x2 y2 Subject Hours Studies (x) Mark xy (y) 350 25 4900 Mathematics, Grade 9 5 70 English, Grade 9 3 65 195 9 4225 Science, Grade 9 4 68 272 16 4624 288 16 5184 Geography, Grade 9 4 72 76 4 1444 French, Grade 9 2 38 Mathematics, Grade 10 7 74 518 49 5476 345 25 4761 English, Grade 10 5 69 426 36 5041 Science, Grade 10 6 71 History, Grade 10 5 75 375 25 5625 912 144 5776 Mathematics, Grade 11 12 76 666 81 5476 English, Grade 11 9 74 1092 196 6084 Physics, Grade 11 14 78 76 830 5515 626 58616 T O T A L Note: Use 4 decimals where applicable. You must do this question by hand. a. Determine the dependent variable. (1 mark) b. Obtain the regression equation. (3 marks)

Mark

M. Dadar Solution

where

a

12(5515)  76(830)  1.785714286 12(626)  (76) 2

76 830  1.785714286( ) 12 12  57.85714286

b

y  1.7857x  57.85714

c. Interpret the meaning of the regression coefficients a and b. (2 marks) In this scenario a represents the amount that mark goes up for every 1hr increase in the amount of time spent studying and b represents Carla’s mark if she did not study at all for the exam. d. Predict her mark for a subject in which she studied 8 hours for the exam. (1 mark) y = 1.7857 (8) + 57.85714 = 72.143 (approx.) e. Compute the coefficient of determination. (3 marks)

r2 

[12(5515)  (76)(830)]2 [12(626)  762 ][12(58616)  8302 ]

M. Dadar Solution

r 2=

3100 2  0. 381984148 1736 14492

f. Compute the correlation coefficient. (1 mark) r  0.381984148  0.618048661 You should use Excel for 10b). 10. A laboratory technician monitors the growth of a bacterial culture by scanning it every hour and estimating the number of bacteria. 0 1 2 3 4 5 6 7 Time (h) 5 9 21 40 82 165 320 614 Population a)

Explain what type of non-linear model would probably best represent this data? (2 marks)

The data appears to be roughly doubling per each hour elapsed so it would be best represented by an exponential model.

b)

Use Excel and display the coefficient of determination and the non-linear regression equation for this data. Copy and paste your solution below. (4 marks) 700 y = 4.9307e0.6956x R² = 0.9992

600 500 400

Series1 300

Expon. (Series1)

200 100 0 0

2

4

6

8

c) Using the model that you obtained in part b), predict how many bacteria will be present after the culture is allowed to grow for a full day. (2 marks) y = 4.9307 e ^(0.6956(24)) = 87739343 so roughly about 88million bacteria will be present after a day.

M. Dadar Solution

Thinking 11.

Does the slope of the line of best fit tell you anything about the strength of a linear correlation? Explain why or why not. (2 Marks)

No, the slope has nothing to do with the strength of line of best fit. The slope is determined based on the data values and its magnitude changes based on the ratio of dependent to independent variable.

Use Fathom for question #12a. A psychologist wants to know if there is a relationship between the number of hours a university student gets per night and his/her overall average. The psychologist collects the following data.

12.

Hours of Sleep Average Mark

6.0 62

6.5 58

7.0 66

6.5 71

8.5 76

8.0 82

9.0 76

8.5 75

7.0 70

7.5 68

6.5 56

a)

Create a scatter plot for this data using Fathom. Display the least squares line and create a residual plot. Copy your solution below. (6 marks)

b)

Use the information from part a) and find the correlation coefficient. Classify the relationship. (2 marks) r = (0.58)^ 0.5 = 0.76

c)

-

Strong Positive Correlation

Find the residual corresponding to the datum (8.0, 82). (2 marks) Y predicted = 6.42 (8) + 22.4 = 73.76

e = 82 – 73.76 = 8.24

7.5 77

M. Dadar Solution Part C: Communication 13. The scatter plot to the right was produced using a TI-83+. The plot shows the relationship between the mass, in grams, of a bacterial culture produced in a laboratory over time, in hours. a) Below are outputs from two regression analyses on the data in the scatter plot. Which analysis best suits this data? Justify your choice. [2 Marks]

In this case the exponential regression is a better estimate since the value of the coefficient of determination, r2, is higher and the scatter plot is curvilinear.

14. Referring to question 12, can we conclude that increasing the number of hours of sleep a student gets will increase the student’s mark? Explain your answer. (2 marks) No. This is not a cause and effect relationship and we cannot conclude that. Even if the correlation coefficient was closer to 1, still we could not say “it will increase the students’ mark” since that would imply causality. Strong correlation does not necessarily mean causality....