Suppose a teacher observed a correlation of 0.38 between age and number of words children could define on a vocabulary test, for a group of nine children aged 7 to 15. The teacher wanted to confirm that there was a correlation between age and vocabulary test scores for the population of children in this age range. She performed a simulation with 1000 randomized orders, scrambling the test scores across the ages, similar to the simulation used in Example 15.3. She was surprised to find that 29% of the simulated samples resulted in correlations of 0.38 or larger, even though there should not have been any correlation between ages and the scrambled test scores.
a. What null and alternative hypotheses was the teacher testing?
b. What is the estimated p-value for her test?
c. Do these results confirm that there is no correlation between age and vocabulary scores for the population of children aged 7 to 15? Explain.
d. Would a simulation with 10,000 randomized orders be much more likely, much less likely, or about equally likely to enable the teacher to reject the null hypothesis, compared to the simulation performed with 1000 randomized orders?
e. The teacher plans to repeat the experiment. What could she do differently to improve the chance of rejecting the null hypothesis?