Hypothesis test of Pearson’s product-moment correlation coefficient

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]

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]

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.

1 A wildlife expert measured the neck lengths, \(x\) metres, and the tail lengths, \(y\) metres, of a sample of 12 mature male giraffes as part of a study into their physical characteristics. The results are shown in the table.

1 A wildlife expert measured the neck lengths, \(x\) metres, and the tail lengths, \(y\) metres, of a sample of 12 mature male giraffes as part of a study into their physical characteristics. The results are shown in the table.

3 A student is investigating the relationship between the length \(x \mathrm {~mm}\) and circumference \(y \mathrm {~mm}\) of plums from a large crop. The student measures the dimensions of a random sample of 10 plums from this crop. Summary statistics for these dimensions are as follows. $$\begin{aligned} & \sum x = 4715 \quad \sum y = 13175 \quad \sum x ^ { 2 } = 2237725 \\ & \sum y ^ { 2 } = 17455825 \quad \sum x y = 6235575 \quad n = 10 \end{aligned}$$

1. Calculate the sample product moment correlation coefficient.
2. Carry out a hypothesis test at the \(5 \%\) significance level to determine whether there is any correlation between length and circumference of plums from this crop. State your hypotheses clearly, defining any symbols which you use.
3. (A) Explain the meaning of a 5\% significance level.
   (B) State one advantage and one disadvantage of using a \(1 \%\) significance level rather than a \(5 \%\) significance level in a hypothesis test. The student decides to take another random sample of 10 plums. Using the same hypotheses as in part (ii), the correlation coefficient for this second sample is significant at the \(5 \%\) level. The student decides to ignore the first result and concludes that there is correlation between the length and circumference of plums in the crop.
4. Comment on the student's decision to ignore the first result. Suggest a better way in which the student could proceed.

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]

3. 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. Laxmi chooses a random sample of 40 adult males and calculates Pearson's product-moment correlation coefficient, \(r\). She finds that \(r = 0.2705\).
2. Use the table below to carry out the test at the \(5 \%\) significance level. Critical values of Pearson's product-moment correlation coefficient.

   \cline{2-5}	1-tail test	\(5 \%\)	\(2.5 \%\)	\(1 \%\)
   2-tail test	\(10 \%\)	\(5 \%\)	\(2.5 \%\)	\(1 \%\)
   38	0.2709	0.3202	0.3760	0.4128
   39	0.2673	0.3160	0.3712	0.4076
   40	0.2638	0.3120	0.3665	0.4026
   41	0.2605	0.3081	0.3621	0.3978

6 A random sample of 15 observations of pairs of values of two variables gives a product moment correlation coefficient of 0.430 .

1. Test at the \(10 \%\) significance level whether there is evidence of non-zero correlation between the variables.
   A second random sample of \(N\) observations gives a product moment correlation coefficient of 0.615 . Using a 5\% significance level, there is evidence of positive correlation between the variables.
2. Find the least possible value of \(N\), justifying your answer.

5 For a random sample of 12 observations of pairs of values \(( x , y )\), the product moment correlation coefficient is - 0.456 . Test, at the \(5 \%\) significance level, whether there is evidence of negative correlation between the variables.

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.

Answer only one of the following two alternatives. EITHER A particle \(P\) of mass \(m\) is attached to one end of a light elastic string of modulus of elasticity \(4mg\) and natural length \(l\). The other end of the string is attached to a fixed point \(O\). The particle rests in equilibrium at the point \(E\), vertically below \(O\). The particle is pulled down a vertical distance \(\frac{3l}{4}\) from \(E\) and released from rest. Show that the motion of \(P\) is simple harmonic with period \(\pi\sqrt{\left(\frac{l}{g}\right)}\). [4] At an instant when \(P\) is moving vertically downwards through \(E\), the string is cut. When \(P\) has descended a further distance \(\frac{5l}{4}\) under gravity, it strikes a fixed smooth plane which is inclined at 30° to the horizontal. The coefficient of restitution between \(P\) and the plane is \(\frac{1}{3}\). Show that the speed of \(P\) immediately after the impact is \(\frac{1}{3}\sqrt{(5gl)}\). [8] OR A new restaurant \(S\) has recently opened in a particular town. In order to investigate any effect of \(S\) on an existing restaurant \(R\), the daily takings, \(x\) and \(y\) in thousands of dollars, at \(R\) and \(S\) respectively are recorded for a random sample of 8 days during a six-month period. The results are shown in the following table.

Day	1	2	3	4	5	6	7	8
\(x\)	1.2	1.4	0.9	1.1	0.8	1.0	0.6	1.5
\(y\)	0.3	0.4	0.6	0.6	0.25	0.75	0.6	0.35

1. Calculate the product moment correlation coefficient for this sample. [4]
2. Stating your hypotheses, test, at the 2.5\% significance level, whether there is negative correlation between daily takings at the two restaurants and comment on your result in the context of the question. [5]

Another sample is taken over \(N\) randomly chosen days and the product moment correlation coefficient is found to be \(-0.431\). A test, at the 5\% significance level, shows that there is evidence of negative correlation between daily takings in the two restaurants.

1. Find the range of possible values of \(N\). [3]

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]

A book collector compared the prices of some books, \(£x\), when new in 1972 and the prices of copies of the same books, \(£y\), on a second-hand website in 2018. The results are shown in Table 1 and are summarised below the table.

Book	A	B	C	D	E	F	G	H	I	J	K	L
\(x\)	0.95	0.65	0.70	0.90	0.55	1.40	1.50	0.50	1.15	0.35	0.20	0.35
\(y\)	6.06	7.00	2.00	5.87	4.00	5.36	7.19	2.50	3.00	8.29	1.37	2.00

Table 1 \(n = 12, \Sigma x = 9.20, \Sigma y = 54.64, \Sigma x^2 = 8.9950, \Sigma y^2 = 310.4572, \Sigma xy = 46.0545\)

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. [1]
   2. Test at the 5\% significance level whether there is evidence of positive correlation between prices in 1972 and prices in 2018. [5]
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. [2]

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.

The following table gives the mean per capita consumption of mozzarella cheese per annum, \(x\) pounds, and the number of civil engineering doctorates awarded, \(y\), in the United States in each of 10 years.

\(x\)	9.3	9.7	9.7	9.7	9.9	10.2	10.5	11.0	10.6	10.6
\(y\)	480	501	540	552	547	622	655	701	712	708

source: www.tylervigen.com

1. Find the equation of the regression line of \(y\) on \(x\). [2]

You are given that the product moment correlation coefficient is 0.959.

1. Explain whether this value would be different if \(x\) is measured in kilograms instead of pounds. [1]

It is desired to carry out a hypothesis test to investigate whether there is correlation between these two variables.

1. Assume that the data is a random sample of all years.
   1. Carry out the test at the 10\% significance level. [6]
   2. Explain whether your conclusion suggests that manufacturers of mozzarella cheese could increase consumption by sponsoring doctoral candidates in civil engineering. [1]

2. A random sample of 8 students sat examinations in Geography and Statistics. The product moment correlation coefficient between their results was 0.572 and the Spearman rank correlation coefficient was 0.655 .

1. Test both of these values for positive correlation. Use a \(5 \%\) level of significance.
2. Comment on your results.

10 A researcher plans to carry out a statistical investigation to test whether there is linear correlation between the time ( \(T\) weeks) from conception to birth, and the birth weight ( \(W\) grams) of new-born babies.

1. Explain why a 1-tail test is appropriate in this context. The researcher records the values of \(T\) and \(W\) for a random sample of 11 babies. They calculate Pearson's product-moment correlation coefficient for the sample and find that the value is 0.722 .
2. Use the table below to carry out the test at the \(1 \%\) significance level. \section*{Critical values of Pearson's product-moment correlation coefficient.}

   \multirow{2}{*}{}	1-tail test	5\%	2.5\%	1\%	0.5\%
   	2-tail test	10\%	5\%	2.5\%	1\%
   \multirow{4}{*}{\(n\)}	10	0.5494	0.6319	0.7155	0.7646
   	11	0.5214	0.6021	0.6851	0.7348
   	12	0.4973	0.5760	0.6581	0.7079
   	13	0.4762	0.5529	0.6339	0.6835