5.08d Hypothesis test: Pearson correlation

6 A pollution control officer is investigating a possible link between the levels of various pollutants in the air and the speed of the wind at various sites. A random sample of 60 values of the windspeed together with the levels of a variety of pollutants is taken at a particular site. The product moment correlation coefficient between wind-speed and nitrogen dioxide level is 0.3231 .

1. Carry out a hypothesis test at the \(10 \%\) significance level to investigate whether there is any correlation between wind-speed and nitrogen dioxide level.
2. State the condition required for the test carried out in part (a) to be valid. Table 6.1 shows the values of the product moment correlation coefficient between 5 different measures of pollution and also wind-speed for a very large random sample of values at another site. Those correlations that are significant at the \(10 \%\) level are denoted by a * after the value of the correlation. \begin{table}[h]

   Correlations	PM10	SPEED	\(\mathrm { NO } _ { 2 }\)	\(\mathrm { O } _ { 3 }\)	PM25	\(\mathrm { SO } _ { 2 }\)
   PM10	1.00
   SPEED	0.08*	1.00
   \(\mathrm { NO } _ { 2 }\)	0.59*	0.25*	1.00
   \(\mathbf { O } _ { \mathbf { 3 } }\)	-0.05*	-0.04*	-0.30*	1.00
   PM25	0.85*	-0.01	0.56*	-0.02	1.00
   \(\mathrm { SO } _ { 2 }\)	0.42*	0.15*	0.73*	-0.63*	0.40*	1.00

   \captionsetup{labelformat=empty} \caption{Table 6.1}
   \end{table} \begin{table}[h]
   \captionsetup{labelformat=empty} \caption{Table 6.2 shows standard guidelines for effect sizes.}

   Product moment correlation coefficient	Effect size
   0.1	Small
   0.3	Medium
   0.5	Large

   \end{table} Table 6.2 The officer analyses these data for effect size.
3. Explain how the very large sample size relates to the interpretation of the correlation coefficients shown in Table 6.1.
4. Comment briefly on what the pollution control officer might conclude from these tables, relevant to her investigation into wind-speed and pollutant levels.

2. An economist suggested the rate of unemployment and the rate of wage inflation are independent. Amy sets about investigating this suggestion. She collects unemployment data and wage inflation data from a random sample of regions in the UK and decides that it is appropriate to carry out a significance test on Pearson's product moment correlation coefficient. Amy's summary statistics for percentage unemployment, \(x\), and percentage wage inflation, \(y\), are shown below. $$\begin{array} { l l l } \sum x = 62 \cdot 8 & \sum y = 19 \cdot 4 & n = 10 \\ \sum x ^ { 2 } = 413 \cdot 44 & \sum y ^ { 2 } = 46 \cdot 16 & \sum x y = 113 \cdot 16 \end{array}$$

1. Calculate Pearson's product moment correlation coefficient for these data.
2. Carry out Amy's test at the \(5 \%\) level of significance and state whether the economist's suggestion is reasonable. Amy also collects unemployment data and wage inflation data from a random sample of 10 regions in Spain and calculates Pearson's product moment correlation coefficient to be - 0.2525 .
3. Should this change Amy's opinion on the economist's suggestion above? What could she do to improve her investigation?
4. What assumption has Amy made in deciding that it is appropriate to carry out a significance test on Pearson's product moment correlation coefficient?

1. An estate agent in Tornep believes that houses further from the railway station are more expensive than those that are closer. She took a random sample of 22 three-bedroom houses in Tornep and calculated the product moment correlation coefficient between the house price and the distance from the station to be 0.3892

Stating your hypotheses clearly, use a \(5 \%\) level of significance to test the estate agent's belief. State the critical region used in your test.

7 The table shows the values of 5 observations of bivariate data \(( x , y )\). $$n = 5 , \Sigma x = 33.1 , \Sigma y = 56.6 , \Sigma x ^ { 2 } = 227.95 , \Sigma y ^ { 2 } = 664.26 , \Sigma x y = 362.37$$

1. Calculate Pearson's product-moment correlation coefficient \(r\) for the data.
2. State what this value of \(r\) tells you about a scatter diagram illustrating the data.
3. Test at the \(5 \%\) significance level whether there is association between \(x\) and \(y\).
4. State the value of Spearman's rank correlation coefficient \(r _ { s }\) for the data.
5. State whether \(r , r _ { s }\), or both or neither is changed when the values of \(x\) are replaced by
   1. \(3 x - 2\),
   2. \(\sqrt { x }\).

5 The birth rate, \(x\) per thousand members of the population, and the life expectancy at birth, \(y\) years, in 14 randomly selected African countries are given in the table. \(n = 14 , \sum x = 72.8 , \sum y = 826 , \sum x ^ { 2 } = 392.96 , \sum y ^ { 2 } = 48924.54 , \sum x y = 4279.16\)

1. Calculate Pearson's product-moment correlation coefficient \(r\) for the data.
2. State what would be the effect on the value of \(r\) if the birth rate were given per hundred and not per thousand.
3. Explain what the sign of \(r\) tells you about the relationship between life expectancy and birth rate for these countries.
4. Test at the \(5 \%\) significance level whether there is correlation between birth rate and life expectancy at birth in African countries.
5. A researcher wants to estimate the life expectancy at birth in Zimbabwe, where the birth rate is 3.9 per thousand. Explain whether a reliable estimate could be obtained using the regression line of \(y\) on \(x\) for the given data.

7 [Figure 1, printed on the insert, is provided for use in this question.]
Harold considers himself to be an expert in assessing the auction value of antiques. He regularly visits car boot sales to buy items that he then sells at his local auction rooms. Harold's father, Albert, who is not convinced of his son's expertise, collects the following data from a random sample of 12 items bought by Harold.

1. Calculate the value of the product moment correlation coefficient between \(x\) and \(y\).
2. Interpret your value in the context of this question.
3. 1. On Figure 1, complete the scatter diagram for these data.
   2. Comment on what this reveals.
4. When items J and L are omitted from the data, it is found that $$S _ { x x } = 4854.4 \quad S _ { y y } = 4216.1 \quad S _ { x y } = 4268.8$$
   1. Calculate the value of the product moment correlation coefficient between \(x\) and \(y\) for the remaining 10 items.
   2. Hence revise as necessary your interpretation in part (b).

3 Fourteen candidates each sat two test papers, Paper 1 and Paper 2, on the same day. The marks, out of a total of 50, achieved by the students on each paper are shown in the table.

4 Stephan is a roofing contractor who is often required to replace loose ridge tiles on house roofs. In order to help him to quote more accurately the prices for such jobs in the future, he records, for each of 11 recently repaired roofs, the number of ridge tiles replaced, \(x _ { i }\), and the time taken, \(y _ { i }\) hours. His results are shown in the table.

1. The pairs of data values for roofs 1 to 7 are plotted on the scatter diagram shown on the opposite page. Plot the 4 pairs of data values for roofs 8 to 11 on the scatter diagram.
2. 1. Calculate the equation of the least squares regression line of \(y _ { i }\) on \(x _ { i }\), and draw your line on the scatter diagram.
   2. Interpret your values for the gradient and for the intercept of this regression line.
3. Estimate the time that it would take Stephan to replace 15 loose ridge tiles on a house roof.
4. Given that \(r _ { i }\) denotes the residual for the point representing roof \(i\) :
   1. calculate the value of \(r _ { 6 }\);
   2. state why the value of \(\sum _ { i = 1 } ^ { 11 } r _ { i }\) gives no useful information about the connection between the number of ridge tiles replaced and the time taken.
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      \section*{Answer space for question 4}
      \includegraphics[max width=\textwidth, alt={}]{6fbb8891-e6de-42fe-a195-ea643552fdcf-11_2385_1714_322_155}

2 The table below shows the heart rates, \(x\) beats per minute, and the systolic blood pressures, \(y\) milligrams of mercury, of a random sample of 10 patients undergoing kidney dialysis.

1. Calculate the value of the product moment correlation coefficient for these data.
2. Assuming that these data come from a bivariate normal distribution, investigate, at the \(1 \%\) level of significance, the claim that, for patients undergoing kidney dialysis, there is a positive correlation between heart rate and systolic blood pressure.

5. A hotel owner in Cardiff is interested in what factors hotel guests think are important when staying at a hotel. From a hotel booking website he collects the ratings for 'Cleanliness', 'Location', 'Comfort' and 'Value for money' for a random sample of 17 Cardiff hotels.
(Each rating is the average of all scores awarded by guests who have contributed reviews using a scale from 1 to 10 , where 10 is 'Excellent'.) The scatter graph shows the relationship between 'Value for money' and 'Cleanliness' for the sample of Cardiff hotels. \includegraphics[max width=\textwidth, alt={}, center]{b35e94ab-a426-4fca-9ecb-c659e0143ed7-4_693_1033_749_516}

1. The product moment correlation coefficient for 'Value for money' and 'Cleanliness' for the sample of 17 Cardiff hotels is 0.895 . Stating your hypotheses clearly, test, at the \(5 \%\) level of significance, whether this correlation is significant. State your conclusion in context.
2. The hotel owner also wishes to investigate whether 'Value for money' has a significant correlation with 'Cost per night'. He used a statistical analysis package which provided the following output which includes the Pearson correlation coefficient of interest and the corresponding \(p\)-value.

   	Value for money	Cost per night
   Value for money	1
   Cost per night	0.047 \(( 0.859 )\)	1

   Comment on the correlation between 'Value for money' and 'Cost per night'.

3 Sixteen candidates took an examination paper in mechanics and an examination paper in statistics.

1. For all sixteen candidates, the value of the product moment correlation coefficient \(r\) for the marks on the two papers was 0.701 correct to 3 significant figures. Test whether there is evidence, at the \(5 \%\) significance level, of association between the marks on the two papers.
2. A teacher decided to omit the marks of the candidates who were in the top three places in mechanics and the candidates who were in the bottom three places in mechanics. The marks for the remaining 10 candidates can be summarised by \(n = 10 , \Sigma x = 750 , \Sigma y = 690 , \Sigma x ^ { 2 } = 57690 , \Sigma y ^ { 2 } = 49676 , \Sigma x y = 50829\).
   1. Calculate the value of \(r\) for these 10 candidates.
   2. What do the two values of \(r\), in parts (a) and (b)(i), tell you about the scores of the sixteen candidates? A bag contains a mixture of blue and green beads, in unknown proportions. The proportion of green beads in the bag is denoted by \(p\).
      1. Sasha selects 10 beads at random, with replacement. Write down an expression, in terms of \(p\), for the variance of the number of green beads Sasha selects. Freda selects one bead at random from the bag, notes its colour, and replaces it in the bag. She continues to select beads in this way until a green bead is selected. The first green bead is the \(X\) th bead that Freda selects.
      2. Assume that \(p = 0.3\). Find
         1. \(\mathrm { P } ( X \geqslant 5 )\),
         2. \(\operatorname { Var } ( X )\).
   3. In fact, on the basis of a large number of observations of \(X\), it is found that \(\mathrm { P } ( X = 3 ) = \frac { 4 } { 25 } \times \mathrm { P } ( X = 1 )\). Estimate the value of \(p\).

2 A book collector compared the prices of some books, \(\pounds x\), when new in 1972 and the prices of copies of the same books, \(\pounds y\), on a second-hand website in 2018.
The results are shown in Table 1 and are summarised below the table. \begin{table}[h] \end{table} $$n = 12 , \Sigma x = 9.20 , \Sigma y = 54.64 , \Sigma x ^ { 2 } = 8.9950 , \Sigma y ^ { 2 } = 310.4572 , \Sigma x y = 46.0545$$

1. It is given that the value of Pearson's product-moment correlation coefficient for the data is 0.381 , correct to 3 significant figures.
   1. State what this information tells you about a scatter diagram illustrating the data.
   2. Test at the \(5 \%\) significance level whether there is evidence of positive correlation between prices in 1972 and prices in 2018.
2. The collector noticed that the second-hand copy of book J was unusually expensive and he decided to ignore the data for book J. Calculate the value of Pearson's product-moment correlation coefficient for the other 11 books.

A random sample of 5 pairs of values \((x, y)\) is given in the following table.

1. Find, showing all necessary working, the equation of the regression line of \(y\) on \(x\). [4]
2. Find, showing all necessary working, the value of the product moment correlation coefficient for this sample. [3]
3. Test, at the 10% significance level, whether there is evidence of non-zero correlation between the variables. [4]

A random sample of 5 pairs of values \((x, y)\) is given in the following table.

\(x\)	1	2	4	5	8
\(y\)	7	5	8	6	4

1. Find, showing all necessary working, the equation of the regression line of \(y\) on \(x\). [4]
2. Find, showing all necessary working, the value of the product moment correlation coefficient for this sample. [3]
3. Test, at the 10% significance level, whether there is evidence of non-zero correlation between the variables. [4]

A random sample of twelve pairs of values of \(x\) and \(y\) is taken from a bivariate distribution. The equations of the regression lines of \(y\) on \(x\) and of \(x\) on \(y\) are respectively $$y = 0.46x + 1.62 \quad \text{and} \quad x = 0.93y + 8.24.$$

1. Find the value of the product moment correlation coefficient for this sample. [2]
2. Using a \(5\%\) significance level, test whether there is non-zero correlation between the variables. [4]

For each month of a certain year, a weather station recorded the average rainfall per day, \(x\) mm, and the average amount of sunshine per day, \(y\) hours. The results are summarised below. \(n = 12\), \(\Sigma x = 24.29\), \(\Sigma x^2 = 50.146\), \(\Sigma y = 45.8\), \(\Sigma y^2 = 211.16\), \(\Sigma xy = 88.415\).

1. Find the mean values, \(\bar{x}\) and \(\bar{y}\). [1]
2. Calculate the gradient of the line of regression of \(y\) on \(x\). [2]
3. Use the answers to parts (i) and (ii) to obtain the equation of the line of regression of \(y\) on \(x\). [2]
4. Find the product moment correlation coefficient and comment, in context, on its value. [4]
5. Stating your hypotheses, test at the 1% level of significance whether there is negative correlation between average rainfall per day and average amount of sunshine per day. [4]

A random sample of 10 pairs of values of \(x\) and \(y\) is given in the following table.

\(x\)	4	6	6	8	2	7	12	14	9	5
\(y\)	2	4	6	8	6	10	9	8	6	5

1. Find the equation of the regression line of \(y\) on \(x\). [4]
2. Find the product moment correlation coefficient for the sample. [2]
3. Find the estimated value of \(y\) when \(x = 10\), and comment on the reliability of this estimate. [2]
4. Another sample of \(N\) pairs of data from the same population has the same product moment correlation coefficient as the first sample given. A test, at the 1% significance level, on this second sample indicates that there is sufficient evidence to conclude that there is positive correlation. Find the set of possible values of \(N\). [3]

A random sample of 8 students is chosen from those sitting examinations in both Mathematics and French. Their marks in Mathematics, \(x\), and in French, \(y\), are summarised as follows. $$\Sigma x = 472 \qquad \Sigma x^2 = 29950 \qquad \Sigma y = 400 \qquad \Sigma y^2 = 21226 \qquad \Sigma xy = 24879$$ Another student scored 72 marks in the Mathematics examination but was unable to sit the French examination. Estimate the mark that this student would have obtained in the French examination. [5] Test, at the 5% significance level, whether there is non-zero correlation between marks in Mathematics and marks in French. [6]

For a random sample of 10 observations of pairs of values \((x, y)\), the equation of the regression line of \(y\) on \(x\) is \(y = 1.1664 + 0.4604x\). It is given that $$\Sigma x^2 = 1419.98 \quad \text{and} \quad \Sigma y^2 = 439.68.$$ The mean value of \(y\) is 6.24.

1. Find the equation of the regression line of \(x\) on \(y\). [6]
2. Find the product moment correlation coefficient. [2]
3. Test at the 5\% significance level whether there is evidence of positive correlation between the two variables. [4]

For a random sample of 5 observations of pairs of values \((x, y)\), the equation of the regression line of \(y\) on \(x\) is \(y = -4.2 + c\) and the equation of the regression line of \(x\) on \(y\) is \(x = 10.8 + dy\), where \(c\) and \(d\) are constants. The product moment correlation coefficient is \(-0.7214\) and the mean value of \(x\) is 7.018. \begin{enumerate}[label=(\roman*)] \item Test at the 5% significance level whether there is evidence of non-zero correlation between the variables. [4] \item Find the values of \(c\) and \(d\). [5] \item Use an appropriate regression line to estimate the value of \(x\) when \(y = 3.5\), and comment on the reliability of your estimate. [2] \end{enumerate]

Over a period of time, researchers took 10 blood samples from one patient with a blood disease. For each sample, they measured the levels of serum magnesium, \(s\) mg/dl, in the blood and the corresponding level of the disease protein, \(d\) mg/dl. The results are shown in the table. [Use \(\sum s^2 = 141.51\), \(\sum d^2 = 1081.74\) and \(\sum sd = 386.32\)]

1. Draw a scatter diagram to represent these data. [3]
2. State what is measured by the product moment correlation coefficient. [1]
3. Calculate \(S_{ss}\), \(S_{dd}\) and \(S_{sd}\). [3]
4. Calculate the value of the product moment correlation coefficient \(r\) between \(s\) and \(d\). [2]
5. Stating your hypotheses clearly, test, at the 1\% significance level, whether or not the correlation coefficient is greater than zero. [3]
6. With reference to your scatter diagram, comment on your result in part (e). [1]

(Total 13 marks)

For one of the activities at a gymnastics competition, 8 gymnasts were awarded marks out of 10 for each of artistic performance and technical ability. The results were as follows.

Gymnast	\(A\)	\(B\)	\(C\)	\(D\)	\(E\)	\(F\)	\(G\)	\(H\)
Technical ability	8.5	8.6	9.5	7.5	6.8	9.1	9.4	9.2
Artistic performance	6.2	7.5	8.2	6.7	6.0	7.2	8.0	9.1

The value of the product moment correlation coefficient for these data is 0.774.

1. Stating your hypotheses clearly and using a 1% level of significance, interpret this value. [5]
2. Calculate the value of the rank correlation coefficient for these data. [6]
3. Stating your hypotheses clearly and using a 1% level of significance, interpret this coefficient. [3]
4. Explain why the rank correlation coefficient might be the better one to use with these data. [2]

The following data was collected for seven cars, showing their engine size, \(x\) litres, and their fuel consumption, \(y\) km per litre, on a long journey. \(\sum x = 12.905\), \(\sum x^2 = 26.8951\), \(\sum y = 108.2\), \(\sum y^2 = 1781.64\), \(\sum xy = 183.176\).

1. Calculate the equation of the regression line of \(x\) on \(y\), expressing your answer in the form \(x = ay + b\). [6 marks]
2. Calculate the product moment correlation coefficient between \(y\) and \(x\) and give a brief interpretation of its value. [4 marks]
3. Use the equation of the regression line to estimate the value of \(x\) when \(y = 12\). State, with a reason, how accurate you would expect this estimate to be. [3 marks]
4. Comment on the use of the line to find values of \(x\) as \(y\) gets very small. [2 marks]

The marks out of 75 obtained by a group of ten students in their first and second Statistics modules were as follows:

Student	\(A\)	\(B\)	\(C\)	\(D\)	\(E\)	\(F\)	\(G\)	\(H\)	\(I\)	\(J\)
Module 1 \((x)\)	\(54\)	\(33\)	\(42\)	\(71\)	\(60\)	\(27\)	\(39\)	\(46\)	\(59\)	\(64\)
Module 2 \((y)\)	\(50\)	\(22\)	\(44\)	\(58\)	\(42\)	\(19\)	\(35\)	\(46\)	\(55\)	\(60\)

1. Find \(\sum x\) and \(\sum y\). [2 marks]

Given that \(\sum x^2 = 26353\) and \(\sum xy = 22991\),

1. obtain the equation of the regression line of \(y\) on \(x\). [5 marks]
2. Estimate the Module 2 result of a student whose mark in Module 1 was (i) 65, (ii) 5. Explain why one of these estimates is less reliable than the other. [4 marks]

The equation of the regression line of \(x\) on \(y\) is \(x = 0.921y + 9.81\).

1. Deduce the product moment correlation coefficient between \(x\) and \(y\), and briefly interpret its value. [4 marks]

A sports scientist wishes to examine the link between resting pulse and fitness. He records the resting pulse, \(p\), of 20 volunteers and the length of time, \(t\) minutes, that each one can run comfortably at 4 metres per second on a treadmill. The results are summarised by $$\Sigma p = 1176, \quad \Sigma t = 511, \quad \Sigma p^2 = 70932, \quad \Sigma t^2 = 19213, \quad \Sigma pt = 27188.$$

1. Calculate the product moment correlation coefficient for these data. [5 marks]
2. Stating your hypotheses clearly, test at the 1\% level of significance whether there is evidence of people with a lower resting pulse having a higher level of fitness as measured by the test. [4 marks]
3. State an assumption necessary to carry out the test in part (b) and comment on its validity in this case. [2 marks]