1 A researcher believes that there is a negative correlation between money spent by the government on education and population growth in various countries. A random sample of 48 countries is selected to investigate this belief. The level of government spending on education \(x\), measured in suitable units, and the annual percentage population growth rate \(y\), are recorded for these countries. Summary statistics for these data are as follows.
$$\Sigma x = 781.3 \quad \Sigma y = 57.8 \quad \Sigma x ^ { 2 } = 14055 \quad \Sigma y ^ { 2 } = 106.3 \quad \Sigma x y = 880.1 \quad n = 48$$
- Calculate the sample product moment correlation coefficient.
- Carry out a hypothesis test at the \(5 \%\) significance level to investigate the researcher's belief. State your hypotheses clearly, defining any symbols which you use.
- State the distributional assumption which is necessary for this test to be valid. Explain briefly how a scatter diagram may be used to check whether this assumption is likely to be valid.
- A student suggests that if the variables are negatively correlated then population growth rates can be reduced by increasing spending on education. Explain why the student may be wrong. Discuss an alternative explanation for the correlation.
- State briefly one advantage and one disadvantage of using a smaller sample size in this investigation.