Hypothesis test for correlation

4 The number, \(x\), of a certain type of sea shell was counted at 60 randomly chosen sites, each one metre square, along the coastline in country \(A\). The number, \(y\), of the same type of sea shell was counted at 50 randomly chosen sites, each one metre square, along the coastline in country \(B\). The results are summarised as follows, where \(\bar{x}\) and \(\bar{y}\) denote the sample means of \(x\) and \(y\) respectively. $$\bar{x} = 29.2 \quad \Sigma(x - \bar{x})^{2} = 4341.6 \quad \bar{y} = 24.4 \quad \Sigma(y - \bar{y})^{2} = 3732.0$$ Find a \(95\%\) confidence interval for the difference between the mean number of sea shells, per square metre, on the coastlines in country \(A\) and in country \(B\).

10 The lengths, \(x \mathrm {~m}\), and masses, \(y \mathrm {~kg}\), of 12 randomly chosen babies born at a particular hospital last year are summarised as follows. $$\Sigma x = 7.50 \quad \Sigma x ^ { 2 } = 4.73 \quad \Sigma y = 38.6 \quad \Sigma y ^ { 2 } = 124.84 \quad \Sigma x y = 24.25$$ Find the value of the product moment correlation coefficient for this sample. Obtain an estimate for the mass of a baby, born last year at the hospital, whose length is 0.64 m . Test, at the \(2 \%\) significance level, whether there is non-zero correlation between the two variables.

9 Arun, Beth and Charlie are investigating whether there is any association between death rate per 1000 and physician density per 1000. They each collect a random sample of size 10. Arun's sample is shown in Fig.9.1. \begin{table}[h] \end{table}

1. Explain whether or not Arun collected his data from the pre-release material, or whether it is not possible to say. Beth and Charlie collected their samples from the pre-release material. Each of them drew a scatter diagram for their samples. The samples and scatter diagrams are shown in Figs. 9.2 and 9.3.

   Beth's sample	death rate per 1000	physician density per 1000
   Sudan	6.7	0.41
   Cambodia	7.4	0.17
   Gabon	6.2	0.36
   Seychelles	7	0.95
   Mexico	5.4	2.25
   Kuwait	2.3	2.58
   Haiti	7.5	0.23
   Maldives	4	1.04
   Nauru	5.9	1.24
   Jordan	3.4	2.34

   \includegraphics[max width=\textwidth, alt={}]{2b9ce212-84e2-4817-be94-98e2adff12a3-08_545_1024_340_918}
   \begin{table}[h]

   Charlie's sample	death rate per 1000	physician density per 1000
   Vanuata	4	0.17
   Solomon Islands	3.8	0.2
   N. Mariana Islands	4.9	0.36
   Nauru	5.9	1.24
   United Kingdom	9.4	2.81
   Portugal	10.6	3.34
   North Macedonia	9.6	2.87
   Faroe Islands	8.8	2.62
   Bulgaria	14.5	3.99
   St. Kitts and Nevis	7.2	2.52

   \captionsetup{labelformat=empty} \caption{Fig. 9.3}
   \end{table} \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Fig. 9.2} \includegraphics[alt={},max width=\textwidth]{2b9ce212-84e2-4817-be94-98e2adff12a3-08_572_899_1400_1041}
   \end{figure} Arun states that Charlie's sample and Beth's sample cannot both be random for the following reasons.
   Kofi collects a sample of 10 African countries and 10 European countries. The scatter diagram for his results is shown in Fig. 9.4. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{2b9ce212-84e2-4817-be94-98e2adff12a3-09_485_903_902_260} \captionsetup{labelformat=empty} \caption{Fig. 9.4}
   \end{figure}
2. On the copy of Fig. 9.4 in the Printed Answer Booklet, use your knowledge of the pre-release material to identify the points representing the 10 European countries, justifying your choice.

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)

Shona calculated four correlation coefficients using data from the Large Data Set. In each case she calculated the correlation coefficient between the masses of the cars and the CO₂ emissions for varying sample sizes. A summary of these calculations, labelled A to D, are listed in the table below.

	Sample size	Correlation coefficient
A	3827	0.088
B	3735	0.246
C	24	0.400
D	1250	-1.183

Shona would like to use calculation A to test whether there is evidence of positive correlation between mass and CO₂ emissions. She finds the critical value for a one-tailed test at the 5% level for a sample of size 3827 is 0.027

1. 1. State appropriate hypotheses for Shona to use in her test. [1 mark]
   2. Determine if there is sufficient evidence to reject the null hypothesis. Fully justify your answer. [1 mark]
2. Shona's teacher tells her to remove calculation D from the table as it is incorrect. Explain how the teacher knew it was incorrect. [1 mark]
3. Before performing calculation B, Shona cleaned the data. She removed all cars from the Large Data Set that had incorrect masses. Using your knowledge of the large data set, explain what was incorrect about the masses which were removed from the calculation. [1 mark]
4. Apart from CO2 and CO emissions, state one other type of emission that Shona could investigate using the Large Data Set. [1 mark]
5. Wesley claims that calculation C shows that a heavier car causes higher CO2 emissions. Give two reasons why Wesley's claim may be incorrect. [2 marks]

A geyser is a hot spring which erupts from time to time. For two geysers, the duration of each eruption, \(x\) minutes, and the waiting time until the next eruption, \(y\) minutes, are recorded.

1. For a random sample of 50 eruptions of the first geyser, the correlation coefficient between \(x\) and \(y\) is 0.758. The critical value for a 2-tailed hypothesis test for correlation at the 5% level is 0.279. Explain whether or not there is evidence of correlation in the population of eruptions. [2]

The scatter diagram in Fig. 9 shows the data from a random sample of 50 eruptions of the second geyser. \includegraphics{figure_9}

1. Stella claims the scatter diagram shows evidence of correlation between duration of eruption and waiting time. Make two comments about Stella's claim. [2]

A survey was conducted into the annual salary offered for 19 different jobs in 2008. The results were as follows, in thousands of pounds.

15	16	18	19	21	36	36	38	41	41
43	47	51	55	56	60	62	64	110

It was decided to undertake a further study to see if self-esteem was correlated with level of annual salary. A random sample of 11 employees was taken and self-esteem was rated on a scale of 1 to 10 with the highest self-esteem being 10. The results were as follows.

Salary in £10 000's	1	2	3	4	5	6	7	8	9	10	11
Self-esteem	4	3	5	1	7	7	8	5	10	7	9