2.05g Hypothesis test using Pearson's r

4 The table shows the latitude, \(x\) (in degrees correct to 3 significant figures), and the average rainfall \(y\) (in cm correct to 3 significant figures) of five European cities. $$\left[ n = 5 , \Sigma x = 265.0 , \Sigma y = 274.6 , \Sigma x ^ { 2 } = 14176.54 , \Sigma y ^ { 2 } = 15162.22 , \Sigma x y = 14464.10 . \right]$$

1. Calculate the product moment correlation coefficient.
2. The values of \(y\) in the table were in fact obtained from measurements in inches and converted into centimetres by multiplying by 2.54 . State what effect it would have had on the value of the product moment correlation coefficient if it had been calculated using inches instead of centimetres.
3. It is required to estimate the annual rainfall at Bergen, where \(x = 60.4\). Calculate the equation of an appropriate line of regression, giving your answer in simplified form, and use it to find the required estimate.

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
   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.

1. Amar is studying the flight of a bird from its nest.

He measures the bird's height above the ground, \(h\) metres, at time \(t\) seconds for 10 values of \(t\) Amar finds the equation of the regression line for the data to be \(h = 38.6 - 1.28 t\)

1. Interpret the gradient of this line. The product moment correlation coefficient between \(h\) and \(t\) is - 0.510
2. Test whether or not there is evidence of a negative correlation between the height above the ground and the time during the flight.
   You should
   • state your hypotheses clearly
   • use a \(5 \%\) level of significance
   • state the critical value used
   Jane draws the following scatter diagram for Amar's data. \includegraphics[max width=\textwidth, alt={}, center]{ab7f7951-e6fe-4853-bb69-8016cf3e796c-06_1024_1033_1135_516}
3. With reference to the scatter diagram, state, giving a reason, whether or not the regression line \(h = 38.6 - 1.28 t\) is an appropriate model for these data. Jane suggests an improved model using the variable \(u = ( t - k ) ^ { 2 }\) where \(k\) is a constant.
   She obtains the equation \(h = 38.1 - 0.78 u\)
4. Choose a suitable value for \(k\) to write Jane's improved model for \(h\) in terms of \(t\) only.

1. A random sample of 15 days is taken from the large data set for Perth in June and July 1987. The scatter diagram in Figure 1 displays the values of two of the variables for these 15 days.

\begin{figure}[h] \end{figure}

1. Describe the correlation. The variable on the \(x\)-axis is Daily Mean Temperature measured in \({ } ^ { \circ } \mathrm { C }\).
2. Using your knowledge of the large data set,
   1. suggest which variable is on the \(y\)-axis,
   2. state the units that are used in the large data set for this variable. Stav believes that there is a correlation between Daily Total Sunshine and Daily Maximum Relative Humidity at Heathrow. He calculates the product moment correlation coefficient between these two variables for a random sample of 30 days and obtains \(r = - 0.377\)
3. Carry out a suitable test to investigate Stav's belief at a \(5 \%\) level of significance. State clearly
   • your hypotheses
   • your critical value
   On a random day at Heathrow the Daily Maximum Relative Humidity was 97\%
4. Comment on the number of hours of sunshine you would expect on that day, giving a reason for your answer.

1. Marc took a random sample of 16 students from a school and for each student recorded

• the number of letters, \(x\), in their last name
• the number of letters, \(y\), in their first name

His results are shown in the scatter diagram on the next page.

1. Describe the correlation between \(x\) and \(y\). Marc suggests that parents with long last names tend to give their children shorter first names.
2. Using the scatter diagram comment on Marc's suggestion, giving a reason for your answer. The results from Marc's random sample of 16 observations are given in the table below.

   \(x\)	3	6	8	7	5	3	11	3	4	5	4	9	7	10	6	6
   \(y\)	7	7	4	4	6	8	5	5	8	4	7	4	5	5	6	3

3. Use your calculator to find the product moment correlation coefficient between \(x\) and \(y\) for these data.
4. Test whether or not there is evidence of a negative correlation between the number of letters in the last name and the number of letters in the first name. You should
   • state your hypotheses clearly
   • use a \(5 \%\) level of significance
   \section*{Question 2 continued.}
   \includegraphics[max width=\textwidth, alt={}]{10736735-3050-43eb-9e76-011ca6fa48b8-05_1125_1337_294_372}
   \section*{Question 2 continued.} \section*{Question 2 continued.}

2 A researcher is investigating the concentration of bacteria and fungi in the air in buildings. The researcher selects a random sample of 12 buildings and measures the concentrations of bacteria, \(x\), and fungi, \(y\), in the air in each building. Both concentrations are measured in the same standard units. Fig. 2 illustrates the data collected. The researcher wishes to test for a relationship between \(x\) and \(y\). \begin{figure}[h] \end{figure}

1. Explain why a test based on the product moment correlation coefficient is likely to be appropriate for these data. Summary statistics for the data are as follows. \(n = 12 \quad \sum x = 18030 \quad \sum y = 15550 \quad \sum x ^ { 2 } = 31458700 \quad \sum y ^ { 2 } = 21980500 \quad \sum x y = 25626800\)
2. In this question you must show detailed reasoning. Calculate the product moment correlation coefficient between \(x\) and \(y\).
3. Carry out a test at the \(5 \%\) significance level based on the product moment correlation coefficient to investigate whether there is any correlation between concentrations of bacteria and fungi.
4. Explain why, in order for proper inference to be undertaken, the sample should be chosen randomly.

3 A student is investigating the link between temperature (in degrees Celsius) and electricity consumption (in Gigawatt-hours) in the country in which he lives. The student has read that there is strong negative correlation between daily mean temperature over the whole country and daily electricity consumption during a year. He wonders if this applies to an individual season. He therefore obtains data on the mean temperature and electricity consumption on ten randomly selected days in the summer. The spreadsheet output below shows the data, together with a scatter diagram to illustrate the data. \includegraphics[max width=\textwidth, alt={}, center]{5be067ff-4668-48d6-8ed2-b8dfa3e678f7-3_798_1593_639_251}

1. Calculate Pearson's product moment correlation coefficient between daily mean temperature and daily electricity consumption. The student decides to carry out a hypothesis test to investigate whether there is negative correlation between daily mean temperature and daily electricity consumption during the summer.
2. Explain why the student decides to carry out a test based on Pearson's product moment correlation coefficient.
3. Show that the test at the \(5 \%\) significance level does not result in the null hypothesis being rejected.
4. The student concludes that there is no correlation between the variables in the summer months. Comment on the student's conclusion.

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}

11 The scatter diagram shows data, taken from the pre-release data set, for several Local Authorities in one region of the UK in 2011. The diagram shows, for each Local Authority, the number of workers who drove to work, and the number of workers who walked to work. \begin{figure}[h] \end{figure}

1. Four students calculated the value of Pearson's product-moment correlation coefficient for the data in the diagram. Their answers were \(0.913,0.124 , - 0.913\) and - 0.124 . One of these values is correct. Without calculation state, with a reason, which is the correct value.
2. Sanjay makes the following statement.
   "The diagram shows that, in any Local Authority, if there are a large number of people who drive to work there will be a large number who walk to work." Give a reason why this statement is incorrect.
3. Rosie makes the following statement.
   "The diagram must be wrong because it shows good positive correlation. If there are more people driving to work, there will be fewer people walking to work, so there would be negative correlation." Explain briefly why Rosie's statement is incorrect.
4. The diagram shows a fairly close relationship between the two variables. One point on the diagram represents a Local Authority where this relationship is less strong than for the others. On the diagram in the Printed Answer Booklet, label this point A.
5. Given that the point A represents a metropolitan borough, suggest a reason why the relationship is less strong for this Local Authority than for the others in the region. The scatter diagram below shows the corresponding data for the same region in 2001. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{2001} \includegraphics[alt={},max width=\textwidth]{6a6316e4-7b2d-4533-988a-4863d79ce668-09_481_885_388_591}
   \end{figure}
6. (a) State a change that has taken place in the metropolitan borough represented by the point A between 2001 and 2011.
   (b) Suggest a possible reason for this change.

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 \(r \neq 0.894\).

1. A company wants to pay its employees according to their performance at work. Last year's performance score \(x\) and annual salary \(y\), in thousands of dollars, were recorded for a random sample of 10 employees of the company.

The performance scores were $$\begin{array} { l l l l l l l l l l } 15 & 24 & 32 & 39 & 41 & 18 & 16 & 22 & 34 & 42 \end{array}$$ (You may use \(\sum x ^ { 2 } = 9011\) )

1. Find the mean and the variance of these performance scores. The corresponding \(y\) values for these 10 employees are summarised by $$\sum y = 306.1 \quad \text { and } \quad \mathrm { S } _ { y y } = 546.3$$
2. Find the mean and the variance of these \(y\) values. The regression line of \(y\) on \(x\) based on this sample is $$y = 12.0 + 0.659 x$$
3. Find the product moment correlation coefficient for these data.
4. State, giving a reason, whether or not the value of the product moment correlation coefficient supports the use of a regression line to model the relationship between performance score and annual salary. The company decides to use this regression model to determine future salaries.
5. Find the proposed annual salary, in dollars, for an employee who has a performance score of 35

2. A large company is analysing how much money it spends on paper in its offices each year. The number of employees in the office, \(x\), and the amount spent on paper in a year, \(p\) (\$ hundreds), in each of 12 randomly selected offices were recorded. The results are summarised in the following statistics. $$\sum x = 93 \quad \mathrm {~S} _ { x x } = 148.25 \quad \sum p = 273 \quad \sum p ^ { 2 } = 6602.72 \quad \sum x p = 2347$$

1. Show that \(\mathrm { S } _ { x p } = 231.25\)
2. Find the product moment correlation coefficient for these data.
3. Find the equation of the regression line of \(p\) on \(x\) in the form \(p = a + b x\)
4. Give an interpretation of the gradient of your regression line. The director of the company wants to reduce the amount spent on paper each year. He wants each office to aim for a model of the form \(p = \frac { 4 } { 5 } a + \frac { 1 } { 2 } b x\), where \(a\) and \(b\) are the values found in part (c). Using the data for the 93 employees from the 12 offices,
5. estimate the percentage saving in the amount spent on paper each year by the company using the director's model.

3. A company owns two petrol stations \(P\) and \(Q\) along a main road. Total daily sales in the same week for \(P ( \pounds p )\) and for \(Q ( \pounds q )\) are summarised in the table below. When these data are coded using \(x = \frac { p - 4365 } { 100 }\) and \(y = \frac { q - 4340 } { 100 }\), $$\Sigma x = 48.1 , \Sigma y = 52.8 , \Sigma x ^ { 2 } = 486.44 , \Sigma y ^ { 2 } = 613.22 \text { and } \Sigma x y = 204.95 .$$

1. Calculate \(S _ { x y } , S _ { x x }\) and \(S _ { y y }\).
2. Calculate, to 3 significant figures, the value of the product moment correlation coefficient between \(x\) and \(y\).
3. 1. Write down the value of the product moment correlation coefficient between \(p\) and \(q\).
   2. Give an interpretation of this value.

1. A meteorologist believes that there is a relationship between the daily mean windspeed, \(w \mathrm { kn }\), and the daily mean temperature, \(t ^ { \circ } \mathrm { C }\). A random sample of 9 consecutive days is taken from past records from a town in the UK in July and the relevant data is given in the table below.

The meteorologist calculated the product moment correlation coefficient for the 9 days and obtained \(r = 0.609\)

1. Explain why a linear regression model based on these data is unreliable on a day when the mean temperature is \(24 ^ { \circ } \mathrm { C }\)
2. State what is measured by the product moment correlation coefficient.
3. Stating your hypotheses clearly test, at the \(5 \%\) significance level, whether or not the product moment correlation coefficient for the population is greater than zero. Using the same 9 days a location from the large data set gave \(\bar { t } = 27.2\) and \(\bar { w } = 3.5\)
4. Using your knowledge of the large data set, suggest, giving your reason, the location that gave rise to these statistics.

2. A researcher believes that there is a linear relationship between daily mean temperature and daily total rainfall. The 7 places in the northern hemisphere from the large data set are used. The mean of the daily mean temperatures, \(t ^ { \circ } \mathrm { C }\), and the mean of the daily total rainfall, \(s \mathrm {~mm}\), for the month of July in 2015 are shown on the scatter diagram below. \includegraphics[max width=\textwidth, alt={}, center]{565bfa73-8095-4242-80b6-cd47aaff6a31-03_844_1339_497_372}

1. With reference to the scatter diagram, explain why a linear regression model may not be suitable for the relationship between \(t\) and s .
   (1) The researcher calculated the product moment correlation coefficient for the 7 places and obtained \(r = 0.658\).
2. Stating your hypotheses clearly, test at the \(10 \%\) level of significance, whether or not the product moment correlation coefficient for the population is greater than zero.
   (3)
3. Using your knowledge of the large data set, suggest the names of the 2 places labelled \(G\) and \(H\).
   (1)
4. Using your knowledge from the large data set, and with reference to the locations of the two places labelled \(G\) and \(H\), give a reason why these places have the highest temperatures in July.
   (2)
5. Suggest how you could make better use of the large data set to investigate the relationship between daily mean temperature and daily total rainfall.
   (1)
   (Total 7 marks)

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'.

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.

1-tailed test significance level	5\%	2.5\%	1\%	0.5\%
2-tailed test significance level	10\%	5\%	2\%	1\%
Critical value	0.549	0.632	0.716	0.765

Determine what Jamal's conclusion to his investigation should be, justifying your answer. [3 marks]

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 [4 marks]

A medical student believes that, in adults, there is a **negative correlation** between the amount of nicotine in their blood stream and their energy level. The student collected data from a random sample of 50 adults. The correlation coefficient between the amount of nicotine in their blood stream and their energy level was \(-0.45\) Carry out a hypothesis test at the 2.5% significance level to determine if this sample provides evidence to support the student's belief. For \(n = 50\), the critical value for a one-tailed test at the 2.5% level for the population correlation coefficient is 0.2787 [4 marks]

Rebecca is a farmer who is monitoring prices for products to use on her farm. She records the prices of two products made from different grains, wheat and oats, at random points in time, to investigate whether there is any correlation. \includegraphics{figure_1} The product moment correlation coefficient for the data is \(0 \cdot 244\). There are 12 data points, and the \(p\)-value is \(0 \cdot 4447\).

1. Comment on the correlation between the prices of Feed Wheat and Feed Oats. [2]

Rebecca also records the prices of two wheat products at random points in time, to investigate whether there is any correlation. \includegraphics{figure_2} The product moment correlation coefficient for the data is \(0 \cdot 653\). There are 12 data points.

1. Stating your hypotheses clearly, test at the 5% level of significance whether there is any evidence of correlation between the prices of these two products. [5]
2. Without referring to the positioning of the points on the graphs, suggest why the product moment correlation coefficient is higher for the second set of data. [1]

Laxmi wishes to test whether there is linear correlation between the mass and the height of adult males.

1. State, with a reason, whether Laxmi should use a 1-tail or a 2-tail test. [1]

Laxmi chooses a random sample of 40 adult males and calculates Pearson's product-moment correlation coefficient, \(r\). She finds that \(r = 0.2705\).

1. Use the table below to carry out the test at the 5% significance level. [5]

Critical values of Pearson's product-moment correlation coefficient.