5.09a Dependent/independent variables

The table shows some of the values of the seasonally adjusted Unemployment Rate (UR), \(x\)\%, and the Consumer Price Index (CPI), \(y\)\%, in the United Kingdom from April 2008 to July 2010.

Date	April 2008	July 2008	October 2008	January 2009	April 2009	July 2009	October 2009	January 2010	April 2010	July 2010
UR, \(x\)\%	5.2	5.7	6.1	6.8	7.5	7.8	7.8	7.9	7.8	7.7
CPI, \(y\)\%	3.0	4.4	4.5	3.0	2.3	1.8	1.5	3.5	3.7	3.1

These data are summarised below. $$n = 10 \quad \sum x = 70.3 \quad \sum x^2 = 503.45 \quad \sum y = 30.8 \quad \sum y^2 = 103.94 \quad \sum xy = 211.9$$

1. Calculate the product moment correlation coefficient, \(r\), for the data, showing that \(-0.6 < r < -0.5\). [3]
2. Karen says "The negative value of \(r\) shows that when the Unemployment Rate increases, it causes the Consumer Price Index to decrease." Give a criticism of this statement. [1]
3. 1. Calculate the equation of the regression line of \(x\) on \(y\). [3]
   2. Use your equation to estimate the value of the Unemployment Rate in a month when the Consumer Price Index is 4.0\%. [2]

Penshop have stores selling stationary in each of 6 towns. The population, \(P\), in tens of thousands and the monthly turnover, \(T\), in thousands of pounds for each of the shops are as recorded below.

Town	Abberton	Bember	Claster	Deller	Edgeton	Figland
\(P\) (0.000's)	3.2	7.6	5.2	9.0	8.1	4.8
\(T\) (£ 000's)	11.1	12.4	13.3	19.3	17.9	11.8

1. Represent these data on a scatter diagram with \(T\) on the vertical axis. [4]
2. 1. Which town's shop might appear to be underachieving given the populations of the towns?
   2. Suggest two other factors that might affect each shop's turnover. [3]

You may assume that $$\Sigma P = 37.9, \quad \Sigma T = 85.8, \quad \Sigma P^2 = 264.69, \quad \Sigma T^2 = 1286, \quad \Sigma PT = 574.25.$$

1. Find the equation of the regression line of \(T\) on \(P\). [7]
2. Estimate the monthly turnover that might be expected if a shop were opened in Gratton, a town with a population of 68 000. [2]
3. Why might the management of Penshop be reluctant to use the regression line to estimate the monthly turnover they could expect if a shop were opened in Haggin, a town with a population of 172 000? [1]

The owner of a mobile burger-bar believes that hot weather reduces his sales. To investigate the effect on his business he collected data on his daily sales, \(£P\), and the maximum temperature, \(T\)°C, on each of 20 days. He then coded the data, using \(x = T - 20\) and \(y = P - 300\), and calculated the summary statistics given below. $$\Sigma x = 57, \quad \Sigma y = 2222, \quad \Sigma x^2 = 401, \quad \Sigma y^2 = 305576, \quad \Sigma xy = 3871.$$

1. Find an equation of the regression line of \(P\) on \(T\). [9 marks]

The owner of the bar doesn't believe it is profitable for him to run the bar if he takes less than £460 in a day.

1. According to your regression line at what maximum daily temperature, to the nearest degree Celsius, does it become unprofitable for him to run the bar? [3 marks]

The Principal of a school believes that more students are absent on days when the temperature is lower. Over a two-week period in December she records the percentage of students who are absent, \(A\%\), and the temperature, \(T°\)C, at 9 am each morning giving these results.

1. Represent these data on a scatter diagram. [4 marks]

You may use $$\Sigma T = 7, \quad \Sigma A = 143.8, \quad \Sigma T^2 = 137, \quad \Sigma A^2 = 2172.66, \quad \Sigma TA = 20.7$$

1. Calculate the product moment correlation coefficient for these data and comment on the Principal's hypothesis. [6 marks]
2. Find an equation of the regression line of \(A\) on \(T\) in the form \(A = p + qT\). [4 marks]
3. Draw the regression line on your scatter diagram. [2 marks]

In a science investigation into energy conservation in the home, a student is collecting data on the time taken for an electric kettle to boil as the volume of water in the kettle is varied. The student's data are shown in the table below, where \(v\) litres is the volume of water in the kettle and \(t\) seconds is the time taken for the kettle to boil (starting with the water at room temperature in each case). Also shown are summary statistics and a scatter diagram on which the regression line of \(t\) on \(v\) is drawn. \(n = 5\), \(\Sigma v = 3.0\), \(\Sigma t = 564\), \(\Sigma v^2 = 2.20\), \(\Sigma vt = 405.2\). \includegraphics{figure_1}

1. Calculate the equation of the regression line of \(t\) on \(v\), giving your answer in the form \(t = a + bv\). [5]
2. Use this equation to predict the time taken for the kettle to boil when the amount of water which it contains is
   1. 0.5 litres,
   2. 1.5 litres.
   Comment on the reliability of each of these predictions. [4]
3. In the equation of the regression line found in part (i), explain the role of the coefficient of \(v\) in the relationship between time taken and volume of water. [2]
4. Calculate the values of the residuals for \(v = 0.8\) and \(v = 1.0\). [4]
5. Explain how, on a scatter diagram with the regression line drawn accurately on it, a residual could be measured and its sign determined. [3]

A baker is aware that the pH of his sourdough, \(y\), and the hydration, \(x\), affect the taste and texture of the final product. The hydration is measured in ml of water per 100 g of flour (ml/100 g). The baker researches how the pH of his sourdough changes as the hydration changes. The results of his research are shown in the diagram below. \includegraphics{figure_5}

1. Describe the relationship between pH and hydration. [2]
2. The equation of the regression line for \(y\) on \(x\) is $$y = 5.4 - 0.02x.$$
   1. Interpret the gradient and intercept of the regression line in this context.
   2. Estimate the pH of the sourdough when the hydration is 20 ml/100 g. Comment on the reliability of this estimate. [4]

A researcher wishes to investigate the relationship between the amount of carbohydrate and the number of calories in different fruits. He compiles a list of 90 different fruits, e.g. apricots, kiwi fruits, raspberries. As he does not have enough time to collect data for each of the 90 different fruits, he decides to select a simple random sample of 14 different fruits from the list. For each fruit selected, he then uses a dieting website to find the number of calories (kcal) and the amount of carbohydrate (g) in a typical adult portion (e.g. a whole apple, a bunch of 10 grapes, half a cup of strawberries). He enters these data into a spreadsheet for analysis.

1. Explain how the random number function on a calculator could be used to select this sample of 14 different fruits. [3]
2. The scatter graph represents 'Number of calories' against 'Carbohydrate' for the sample of 14 different fruits.
   1. Describe the correlation between 'Number of calories' and 'Carbohydrate'. [1]
   2. Interpret the correlation between 'Number of calories' and 'Carbohydrate' in this context. [1]
   \includegraphics{figure_1}
3. The equation of the regression line for this dataset is: 'Number of calories' = 12.4 + 2.9 × 'Carbohydrate'
   1. Interpret the gradient of the regression line in this context. [1]
   2. Explain why it is reasonable for the regression line to have a non-zero intercept in this context. [1]

A university professor conducted some research into factors that affect job satisfaction. The four factors considered were Interesting work, Good wages, Job security and Appreciation of work done. The professor interviewed workers at 14 different companies and asked them to rate their companies on each of the factors. The workers' ratings were averaged to give each company a score out of 5 on each factor. Each company was also given a score out of 100 for Job satisfaction. The following graph shows the part of the research concerning Job Satisfaction versus Interesting work. \includegraphics{figure_2}

1. Calculate the equation of the least squares regression line of Job satisfaction (\(y\)) on Interesting work (\(x\)), given the following summary statistics. [5] \(\sum x = 46 \cdot 2\), \quad \(\sum y = 898\), \quad \(S_{xx} = 3 \cdot 48\) \(S_{xy} = 49 \cdot 45\), \quad \(S_{yy} = 1437 \cdot 714\), \quad \(n = 14\)
2. Give two reasons why it would be inappropriate for the professor to use this equation to calculate the score for Interesting work from a Job satisfaction score of 90. [2]

For a set of 30 pairs of observations of the variables \(x\) and \(y\), it is known that \(\sum x = 420\) and \(\sum y = 240\). The least squares regression line of \(y\) on \(x\) passes through the point with coordinates \((19, 20)\).

1. Show that the equation of the regression line of \(y\) on \(x\) is \(y = 2 \cdot 4x - 25 \cdot 6\) and use it to predict the value of \(y\) when \(x = 26\). [6]
2. State two reasons why your prediction in part (a) may not be reliable. [2]

A year 12 student wishes to study at a Welsh university. For a randomly chosen year between 2000 and 2017 she collected data for seven universities in Wales from the Complete University Guide website. The data are for the variables: • 'Entry standards' – the average UCAS tariff score of new undergraduate students; • 'Student satisfaction' – a measure of student views of the teaching quality at the university taken from the National Student Survey (maximum 5); • 'Graduate prospects' – a measure of the employability of a university's first degree graduates (maximum 100); • 'Research quality' – a measure of the quality of the research undertaken in the university (maximum 4).

1. Pearson's product-moment correlation coefficients, for each pairing of the four variables, are shown in the table below. Discuss the correlation between graduate prospects and the other three variables. [2]

   Variable	Entry standards	Student satisfaction	Graduate prospects	Research quality
   Entry standards	1
   Student satisfaction	-0.030	1
   Graduate prospects	0.772	0.236	1
   Research quality	0.866	0.066	0.827	1

2. Calculate the equation of the least squares regression line to predict 'Entry standards'(y) from 'Research quality'(x), given the summary statistics: $$\sum x = 22.24, \sum y = 2522, S_{xx} = 1.0542, S_{xy} = 20193.5, S_{yy} = 122.72.$$ [5]
3. The data for one of the Welsh universities are missing. This university has a research quality of 3.00. Use your equation to predict the entry standard for this university. [2]

A large field of wheat is split into 8 plots of equal area. Each plot is treated with a different amount of fertiliser, \(f\) grams/m². The yield of wheat, \(w\) tonnes, from each plot is recorded. The results are summarised below. $$\sum f = 28 \quad \sum w = 303 \quad \sum w^2 = 13447 \quad S_{ff} = 42 \quad S_{fw} = 269.5$$

1. Calculate the product moment correlation coefficient between \(f\) and \(w\) [2]
2. Interpret the value of your product moment correlation coefficient. [1]
3. Find the equation of the regression line of \(w\) on \(f\) in the form \(w = a + bf\) [3]
4. Using your equation, estimate the decrease in yield when the amount of fertiliser decreases by 0.5 grams/m² [1]

The table below shows the typical stopping distances \(d\) metres for a particular car travelling at \(v\) miles per hour.

\(v\)	20	30	40	50	60	70
\(d\)	13	24	36	52	72	94

1. State each of the following words that describe the variable \(v\). Independent \quad Dependent \quad Controlled \quad Response [1]
2. Calculate the equation of the regression line of \(d\) on \(v\). [2]
3. Use the equation found in part (ii) to estimate the typical stopping distance when this car is travelling at 45 miles per hour. [1]

It is given that the product moment correlation coefficient for the data is 0.990 correct to three significant figures.

1. Explain whether your estimate found in part (iii) is reliable. [2]

As part of a study into the effects of alcohol, volunteers have their reaction times measured after they have consumed various fixed amounts of alcohol. For a random sample of 12 volunteers the following information was collected.

Units of alcohol consumed	2	3	3	4	4.5	5.5	6	6	7	8	8	9
Reaction time (seconds)	1	2	5	5	3.8	5.5	4.8	8.5	7.2	6.8	9	8

1. Which is the independent variable in this experiment? [1]
2. Find the least squares regression line of \(y\) (Reaction time) on \(x\) (Units of alcohol), and use it to estimate the reaction time of someone who has consumed 5 units of alcohol. [5]