One-tailed test for positive correlation

9 The land areas \(x\) (in suitable units) and populations \(y\) (in millions) for a sample of 8 randomly chosen cities are given in the following table. $$\left[ \Sigma x = 35.9 , \Sigma x ^ { 2 } = 216.47 , \Sigma y = 36.8 , \Sigma y ^ { 2 } = 244.96 , \Sigma x y = 212.62 . \right]$$

1. Find, showing all necessary working, the value of the product moment correlation coefficient for this sample.
2. Using a \(1 \%\) significance level, test whether there is positive correlation between land area and population of cities.
   The land areas and populations for another randomly chosen sample of cities, this time of size \(n\), give a product moment correlation coefficient of 0.651 . Using a test at the \(1 \%\) significance level, there is evidence of non-zero correlation between the variables.
3. Find the least possible value of \(n\), justifying your answer.

3. Barbara is investigating the relationship between average income (GDP per capita), \(x\) US dollars, and average annual carbon dioxide ( \(\mathrm { CO } _ { 2 }\) ) emissions, \(y\) tonnes, for different countries. She takes a random sample of 24 countries and finds the product moment correlation coefficient between average annual \(\mathrm { CO } _ { 2 }\) emissions and average income to be 0.446

1. Stating your hypotheses clearly, test, at the \(5 \%\) level of significance, whether or not the product moment correlation coefficient for all countries is greater than zero. Barbara believes that a non-linear model would be a better fit to the data.
   She codes the data using the coding \(m = \log _ { 10 } x\) and \(c = \log _ { 10 } y\) and obtains the model \(c = - 1.82 + 0.89 m\) The product moment correlation coefficient between \(c\) and \(m\) is found to be 0.882
2. Explain how this value supports Barbara's belief.
3. Show that the relationship between \(y\) and \(x\) can be written in the form \(y = a x ^ { n }\) where \(a\) and \(n\) are constants to be found.

6. Anna is investigating the relationship between exercise and resting heart rate. She takes a random sample of 19 people in her year at school and records for each person

• their resting heart rate, \(h\) beats per minute
• the number of minutes, \(m\), spent exercising each week

Her results are shown on the scatter diagram. \includegraphics[max width=\textwidth, alt={}, center]{3a09f809-fa28-4b3d-bb69-ea074433bd8f-16_531_551_653_740}

1. Interpret the nature of the relationship between \(h\) and \(m\) Anna codes the data using the formulae $$\begin{aligned} & x = \log _ { 10 } m \\ & y = \log _ { 10 } h \end{aligned}$$ The product moment correlation coefficient between \(x\) and \(y\) is - 0.897
2. Test whether or not there is significant evidence of a negative correlation between \(x\) and \(y\) You should
   • state your hypotheses clearly
   • use a \(5 \%\) level of significance
   • state the critical value used
   The equation of the line of best fit of \(y\) on \(x\) is $$y = - 0.05 x + 1.92$$
3. Use the equation of the line of best fit of \(y\) on \(x\) to find a model for \(h\) on \(m\) in the form $$h = a m ^ { k }$$ where \(a\) and \(k\) are constants to be found.

5 A psychologist investigates the relationship between 'openness' and 'creativity' in adults. Each member of a random sample of 15 adults is given two tests, one on openness and one on creativity. Each test has a maximum score of 75 . The results are given in the table. \(n = 15 \quad \sum x = 519 \quad \sum y = 645 \quad \sum x ^ { 2 } = 19033 \quad \sum y ^ { 2 } = 29751 \quad \sum x y = 23034\)

1. Use Pearson's product-moment correlation coefficient to test, at the \(5 \%\) significance level, whether there is positive association between openness and creativity.
2. State what the value of Pearson's product-moment correlation coefficient shows about a scatter diagram illustrating the data.
3. A student suggests that there is a way to obtain a more accurate measure of the correlation. Before carrying out the test it would be better to standardise the test scores so that they have the same mean and variance. Explain whether you agree with this suggestion.

2 A shopper estimates the cost, \(\pounds X\) per item, of each of 12 items in a supermarket. The shopper's estimates are compared with the actual cost, \(\pounds Y\) per item, of each item. The results are summarised as follows. \(n = 12\) \(\sum x = 399\) \(\sum y = 623.88\) \(\sum x ^ { 2 } = 28127\) \(\sum y ^ { 2 } = 116509.0212\) \(\sum x y = 45006.01\) Test at the 1\% significance level whether the shopper's estimates are positively correlated with the actual cost of the items.

8. The heights, in metres, and weights, in kilograms, of a random sample of 9 men are shown in the table below

1. Given that \(\mathrm { S } _ { x x } = 0.0632 , \mathrm {~S} _ { y y } = 1957.5556\) and \(\mathrm { S } _ { x y } = 9.3433\) calculate, to 3 decimal places, the product moment correlation coefficient between height and weight for these men.
2. Use your value of the product moment correlation coefficient to test whether or not there is evidence of a positive correlation between the height and weight of men. Use a \(5 \%\) significance level. State your hypotheses clearly. Peter does not know the heights or weights of the 9 men. He is given photographs of them and asked to put them in order of increasing weight. He puts them in the order $$A C E B G D I F H$$
3. Find, to 3 decimal places, Spearman's rank correlation coefficient between Peter's order and the actual order.
4. Use your value of Spearman’s rank correlation coefficient to test for evidence of Peter's ability to correctly order men, by their weight, from their photographs. Use a 5\% significance level and state your hypotheses clearly.

1. Phil measures the concentration of a radioactive element, \(c\), and the amount of dissolved solids, \(a\), of 8 random samples of groundwater. His results are shown in the table below.

Given that $$\mathrm { S } _ { c c } = 34787.5 \quad \mathrm {~S} _ { a a } = 0.2172875 \quad \mathrm {~S} _ { c a } = 47.7625$$

1. calculate, to 3 decimal places, the product moment correlation coefficient between the concentration of the radioactive element and the amount of dissolved solids for these groundwater samples.
2. Use your value of the product moment correlation coefficient to test whether or not there is evidence of a positive correlation between the concentration of this radioactive element and the amount of dissolved solids in groundwater. Use a \(5 \%\) significance level. State your hypotheses clearly.
3. Calculate, to 3 decimal places, Spearman's rank correlation coefficient between the concentration of the radioactive element and the amount of dissolved solids.
4. Use your value of Spearman's rank correlation coefficient to test for evidence of a positive correlation between the concentration of the radioactive element and the amount of dissolved solids. Use a \(5 \%\) significance level. State your hypotheses clearly.
5. Using your conclusions in part (b) and part (d), comment on the possible relationship between these variables.

1 The best performances of a random sample of 20 junior athletes in the long jump, \(x\) metres, and in the high jump, \(y\) metres, were recorded. The following statistics were calculated from the results. $$S _ { x x } = 7.0036 \quad S _ { y y } = 0.8464 \quad S _ { x y } = 1.3781$$

1. Calculate the value of the product moment correlation coefficient between \(x\) and \(y\).
   (2 marks)
2. Assuming that these data come from a bivariate normal distribution, investigate, at the \(1 \%\) level of significance, the claim that for junior athletes there is a positive correlation between \(x\) and \(y\).
3. Interpret your conclusion in the context of this question.

2. A Geography teacher is interested in the link between mathematical ability and the ability to visualise three-dimensional situations. He gives a group of 15 students a test and records each student's score, \(m\), on the mathematics questions and each student's score, \(v\), on the visiospatial questions. He calculates the following summary statistics: $$S _ { m m } = 3747.73 , \quad S _ { v v } = 2791.33 , \quad S _ { m v } = 2564.33$$

1. Calculate the product moment correlation coefficient for these data.
2. Stating your hypotheses clearly and using a \(5 \%\) level of significance test the theory that students who are good at Mathematics tend to have better visio-spatial awareness.
   (4 marks)

5 Two practice GCSE examinations in mathematics are given to all of the students in a large year group. A teacher wants to check whether there is a positive relationship between the marks obtained by the students in the two examinations. She selects a random sample of 20 students. Summary data for the marks obtained in the first and second practice examinations, \(x\) and \(y\) respectively, are as follows. $$\sum x = 565 \quad \sum y = 724 \quad \sum x ^ { 2 } = 17103 \quad \sum y ^ { 2 } = 29286 \quad \sum x y = 21635$$ The teacher decides to carry out a hypothesis test based on Pearson’s product moment correlation coefficient.

1. In this question you must show detailed reasoning. Calculate the value of Pearson's product moment correlation coefficient.
2. Carry out the test at the \(5 \%\) significance level.
3. Given that the teacher did not draw a scatter diagram before carrying out the test, comment on the validity of the test.

3 The scatter diagram below illustrates data concerning average annual income per person, \(\\) x\(, and average life expectancy, \)y$ years, for 45 randomly selected cities. \includegraphics[max width=\textwidth, alt={}, center]{464c80be-007b-4d5a-9fe5-2f35100bdea6-3_860_1465_354_244}

1. State whether neither variable, one variable or both variables can be considered to be random in this situation. A student is researching possible positive association between average annual income and average life expectancy. The student decides that the data point labelled A on the scatter diagram is an outlier.
2. Describe the apparent relationship between average annual income and average life expectancy for this data point relative to the rest of the data. The data for point A is removed. The student now wishes to carry out a hypothesis test using the product moment correlation coefficient for the remaining 44 data points to investigate whether there is positive correlation between average annual income and average life expectancy.
3. Explain why this type of hypothesis test is appropriate in this situation. Justify your answer. The summary statistics for these 44 data points are as follows. \(\sum x = 751120 \sum y = 2397.1 \sum x ^ { 2 } = 14363849200 \sum y ^ { 2 } = 133014.63 \sum x y = 42465962\)
4. Determine the value of the product moment correlation coefficient.
5. Carry out the test at the 1\% significance level.

3. Awena has a large data set of body measurements, and she wants to investigate relationships between body dimensions. In this particular investigation, she is testing for a correlation between forearm girth and bicep girth. The diagrams below show how to measure these. \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure}

1. Why is it appropriate for Awena to use a one-tailed test?
   Awena takes a random sample of size 11 from her data set and plots the following scatter diagram. \includegraphics[max width=\textwidth, alt={}, center]{8f47b2ff-f954-42ec-8ecc-fc64313a7b89-07_937_1431_420_312}
2. Using the computer output above, carry out a one-tailed significance test on the sample product moment correlation coefficient at the \(0 \cdot 5 \%\) level.
3. Blodwen also has access to the same large data set. She decides to do the same test using all of the 507 available data points. Her results are shown below. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Forearm girth versus Bicep girth} \includegraphics[alt={},max width=\textwidth]{8f47b2ff-f954-42ec-8ecc-fc64313a7b89-08_933_1504_477_276}
   \end{figure}
   1. State the problem Blodwen will encounter when attempting to use statistical tables for her test.
   2. How should Blodwen deal with this problem?
      \section*{PLEASE DO NOT WRITE ON THIS PAGE}

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.

11 In an experiment involving a bivariate distribution ( \(X , Y\) ) a random sample of 7 pairs of values was obtained and Pearson's product-moment correlation coefficient \(r\) was calculated for these values.

1. The value of \(r\) was found to be 0.894 . Use the table below to test, at the \(5 \%\) significance level, whether there is positive linear correlation in the population, stating your hypotheses and conclusion clearly.

   1-tail test 2-tail test	5\%	2.5\%	1\%	0.5\%
   	10\%	5\%	2\%	1\%
   \(n\)
   1	-	-	-	-
   2	-	-	-	-
   3	0.9877	0.9969	0.9995	0.9999
   4	0.9000	0.9500	0.9800	0.9900
   5	0.8054	0.8783	0.9343	0.9587
   6	0.7293	0.8114	0.8822	0.9587
   7	0.6694	0.7545	0.8329	0.9745
   8	0.6215	0.7067	0.7887	0.8343
   9	0.5882	0.6664	0.7498	0.7977
   10	0.5494	0.6319	0.7155	0.7646

   Scatter diagrams for four sets of bivariate data, are shown. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{85de9a39-f8be-40ee-b0c8-e2e632be93d8-8_380_371_301_191} \captionsetup{labelformat=empty} \caption{Diagram A}
   \end{figure} \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{85de9a39-f8be-40ee-b0c8-e2e632be93d8-8_373_373_301_628} \captionsetup{labelformat=empty} \caption{Diagram B}
   \end{figure} \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{85de9a39-f8be-40ee-b0c8-e2e632be93d8-8_378_373_301_1064} \captionsetup{labelformat=empty} \caption{Diagram C}
   \end{figure} \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{85de9a39-f8be-40ee-b0c8-e2e632be93d8-8_378_373_301_1503} \captionsetup{labelformat=empty} \caption{Diagram D}
   \end{figure} It is given that \(r = 0.894\) for one of these diagrams.
2. For each of the other diagrams, state how you can tell that \(r \neq 0.894\).

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.

15 Jamal, a farmer, claims that the larger the rainfall, the greater the yield of wheat from his farm. He decides to investigate his claim, at the \(5 \%\) level of significance.
He measures the rainfall in centimetres and the yield in kilograms for a random sample of ten years. He correctly calculates the product moment correlation coefficient between rainfall and yield for his sample to be 0.567 The table below shows the critical values for correlation coefficients for a sample size of 10 for different significance levels, for both 1- and 2-tailed tests. Determine what Jamal's conclusion to his investigation should be, justifying your answer. \includegraphics[max width=\textwidth, alt={}, center]{c8a41c47-bbda-4e91-a7a2-d0bcf6a46f25-21_2488_1716_219_153}

16 An educational expert found that the correlation coefficient between the hours of revision and the scores achieved by 25 students in their A-level exams was 0.379 Her data came from a bivariate normal distribution.
Carry out a hypothesis test at the \(1 \%\) significance level to determine if there is a positive correlation between the hours of revision and the scores achieved by students in their A-level exams. The critical value of the correlation coefficient is 0.4622
[0pt] [4 marks]