4 A student is investigating correlations between various personality traits, two of which are conscientiousness and openness to new experiences.
She selects a random sample of 10 students at her university and uses standard tests to measure their conscientiousness and their openness. The product moment correlation coefficient between these two variables for the 10 students is 0.476 .

1. Assuming that the underlying population has a bivariate Normal distribution, carry out a hypothesis test at the \(10 \%\) significance level to investigate whether there is any correlation between openness and conscientiousness in students. Table 4.1 below shows the values of the product moment correlation coefficients between 5 different personality traits for a much larger sample of students. Those correlations that are significant at the \(5 \%\) level are denoted by a * after the value of the correlation. \begin{table}[h]

   	Neuroticism	Extroversion	Openness	Agreeableness	Conscientiousness
   Neuroticism	1
   Extroversion	-0.296*	1
   Openness	-0.044	0.405*	1
   Agreeableness	-0.190*	0.061	0.042	1
   Conscientiousness	-0.485*	0.145	0.235*	0.112	1

   \captionsetup{labelformat=empty} \caption{Table 4.1}
   \end{table} The student analyses these factors for effect size.
   Guidelines often used when considering effect size are given in Table 4.2 below. \begin{table}[h]

   Product moment correlation coefficient	Effect size
   0.1	Small
   0.3	Medium
   0.5	Large

   \captionsetup{labelformat=empty} \caption{Table 4.2}
   \end{table}
2. The student notes that, despite the result of the test in part (a), the correlation between openness and conscientiousness is significant at the \(5 \%\) level with this second sample. Comment briefly on why this may be the case.
3. The student intends to summarise her findings about relationships between these factors, including effect sizes, in a report.
   Use the information in Tables 4.1 and 4.2 to identify two summary points the student could make.