Hypothesis test for correlation

4 Th m b r, \(x , 6\) a certain ty sea sh ll was co ed at \(\theta\) rach ly cb en sites, each \(\mathbf { a }\) metre seq re, alg th co stlie in co ry \(A\). Th \(m \quad \mathbf { b } , y , \boldsymbol { 6 }\) th same \(\quad \mathbf { 6 }\) sea sh ll was co ed at \(\theta\) rach ly cb en sites, each \(\mathbf {} { } _ { \text {t } }\) metre sq re, alog th co stlie in co ry \(B\). Tb results are sm marised as fb lw s,w b re \(\bar { x }\) ad \(\bar { y } \mathbf { d } \mathbf { h }\) e th samp e meas \(\mathbf { 6 } x\) ad \(y\) resp ctiv ly. $$\bar { x } = 9 \quad \Sigma ( x - \bar { x } ) ^ { 2 } = \mathbf { 3 } \quad \bar { y } = \mathbf { 4 } \quad \Sigma ( y - \bar { y } ) ^ { 2 } = \text { 日 }$$ \includegraphics[max width=\textwidth, alt={}]{0df58f9d-6700-46cc-bcf0-903e94cccc02-06_58_1667_539_239} metre,it b co stlin sirc \(\mathbf { b }\) ry \(A\) ad inc \(\mathbf { b }\) ry \(B\).

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.
   • Both samples include Nauru - there should not be any common values.
   • Beth's diagram suggests a negative association between death rate and physician density, whereas Charlie's diagram suggests a positive association. If both samples are random the same relationship would be suggested.
   • - Explain whether Arun’s reasons are valid.
   • State whether or not Arun is correct, or whether it is not possible to say.
   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.

6. The Principal of a school believes that more students are absent on days when the temperature is lower. Over a two-week period in December she records the percentage of students who are absent, \(A \%\), and the temperature, \(T ^ { \circ } \mathrm { C }\), at 9 am each morning giving these results.

1. Represent these data on a scatter diagram. You may use $$\Sigma T = 7 , \quad \Sigma A = 143.8 , \quad \Sigma T ^ { 2 } = 137 , \quad \Sigma A ^ { 2 } = 2172.66 , \quad \Sigma T A = 20.7$$
2. Calculate the product moment correlation coefficient for these data and comment on the Principal’s hypothesis.
3. Find an equation of the regression line of \(A\) on \(T\) in the form \(A = p + q T\).
4. Draw the regression line on your scatter diagram.

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.
\(n = 12\)
\(\sum x ^ { 2 } = 28127\)
\(\sum x = 399\)
\(\Sigma y ^ { 2 } = 116509.0212\)
\(\Sigma y = 623.88\)
\(\sum x y = 45006.01\) Test at the \(1 \%\) significance level whether the shopper's estimates are positively correlated with the actual cost of the items.
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