9.1 Least squares, calculator work, correlation coefficients  (Page 2/2)

1. The table below lists the exam results for 5 students in the subjects of Science and Biology.

   Learner	1	2	3	4	5
   Science %	55	66	74	92	47
   Biology %	48	59	68	84	53

   1. Use the formulae to find the regression equation coefficients $a$ and $b$ .
   2. Draw a scatter plot of the data on graph paper.
   3. Now use algebra to find a more accurate equation.
2. Footlengths and heights of 7 students are given in the table below.

   Height (cm)	170	163	131	181	146	134	166
   Footlength (cm)	27	23	20	28	22	20	24

   1. Draw a scatter plot of the data on graph paper.
   2. Indentify and describe any trends shown in the scatter plot.
   3. Find the equation of the least squares line by using algebraic methods and draw the line on your graph.
   4. Use your equation to predict the height of a student with footlength 21,6 cm.
   5. Use your equation to predict the footlength of a student 176 cm tall.
3. Repeat the data in question 2 and find the regression line using a calculator

Correlation coefficients

Once we have applied regression analysis to a set of data, we would like to have a number that tells us exactly how well the data fits the function. A correlation coefficient, $r$ , is a tool that tells us to what degree there is a relationship between two sets of data. The correlation coefficient $r\in \left[-,1,;,1\right]$ when $r=-1$ , there is a perfect negative relationship, when $r=0$ , there is no relationship and $r=1$ is a perfect positive correlation.

Positive, strong	Positive, fairly strong	Positive, weak	No association	Negative, fairly strong
$r\approx 0,9$	$r\approx 0,7$	$r\approx 0,4$	$r=0$	$r\approx -0,7$

We often use the correlation coefficient $r^2$ in order to examine the strength of the correlation only.

In this case:

The correlation coefficient $r$ can be calculated using the formula

• where $n$ is the number of data points,
• $s_x$ is the standard deviation of the $x$ -values and
• $s_y$ is the standard deviation of the $y$ -values.

This is known as the Pearson's product moment correlation coefficient. It is a long calculation and much easier to do on the calculator where you simply follow the procedure for the regression equation, and go on to find $r$ .

Exercises

1. Below is a list of data concerning 12 countries and their respective carbon dioxide (CO ${}_2$ ) emmission levels per person and the gross domestic product (GDP - a measure of products produced and services delivered within a country in a year) per person.

   	CO ${}_2$ emmissions per capita ( $x$ )	GDP per capita ( $y$ )
   South Africa	8,1	3 938
   Thailand	2,5	2 712
   Italy	7,3	20 943
   Australia	17,0	23 893
   China	2,5	816
   India	0,9	463
   Canada	16,0	22 537
   United Kingdom	9,0	21 785
   United States	19,9	31 806
   Saudi Arabia	11,0	6 853
   Iran	3,8	1 493
   Indonesia	1,2	986

   1. Draw a scatter plot of the data set and your estimate of a line of best fit.
   2. Calculate equation of the line of regression using the method of least squares.
   3. Draw the regression line equation onto the graph.
   4. Calculate the correlation coefficient $r$ .
   5. What conclusion can you reach, regarding the relationship between CO ${}_2$ emission and GDP per capita for the countries in the data set?
2. A collection of data on the peculiar investigation into a foot size and height of students was recorded in the table below. Answer the questions to follow.

   Length of right foot (cm)	Height (cm)
   25,5	163,3
   26,1	164,9
   23,7	165,5
   26,4	173,7
   27,5	174,4
   24	156
   22,6	155,3
   27,1	169,3

   1. Draw a scatter plot of the data set and your estimate of a line of best fit.
   2. Calculate equation of the line of regression using the method of least squares or your calculator.
   3. Draw the regression line equation onto the graph.
   4. Calculate the correlation coefficient $r$ .
   5. What conclusion can you reach, regarding the relationship between the length of the right foot and height of the students in the data set?
3. A class wrote two tests, and the marks for each were recorded in the table below. Full marks in the first test was 50, and the second test was out of 30.
   1. Is there a strong association between the marks for the first and second test? Show why or why not.
   2. One of the learners (in row 18) did not write the second test. Given their mark for the first test, calculate an expected mark for the second test.

   Learner	Test 1	Test 2
   	(Full marks: 50)	(Full marks: 30)
   1	42	25
   2	32	19
   3	31	20
   4	42	26
   5	35	23
   6	23	14
   7	43	24
   8	23	12
   9	24	14
   10	15	10
   11	19	11
   12	13	10
   13	36	22
   14	29	17
   15	29	17
   16	25	16
   17	29	18
   18	17
   19	30	19
   20	28	17

4. A fast food company produces hamburgers. The number of hamburgers made, and the costs are recorded over a week.

   Hamburgers made	Costs
   495	R2382
   550	R2442
   515	R2484
   500	R2400
   480	R2370
   530	R2448
   585	R2805

   1. Find the linear regression function that best fits the data.
   2. If the total cost in a day is R2500, estimate the number of hamburgers produced.
   3. What is the cost of 490 hamburgers?
5. The profits of a new shop are recorded over the first 6 months. The owner wants to predict his future sales. The profits so far have been R90 000 , R93 000, R99 500, R102 000, R101 300, R109 000.
   1. For the profit data, calculate the linear regression function.
   2. Give an estimate of the profits for the next two months.
   3. The owner wants a profit of R130 000. Estimate how many months this will take.
6. A company produces sweets using a machine which runs for a few hours per day. The number of hours running the machine and the number of sweets produced are recorded.

   Machine hours	Sweets produced
   3,80	275
   4,23	287
   4,37	291
   4,10	281
   4,17	286

   Find the linear regression equation for the data, and estimate the machine hours needed to make 300 sweets.