Introduction

Researchers are concerned that exposure to lead pollution may reduce brain development in children. They collect a sample of 121 children who live within a 4 mile radius of a lead smelter and measure their IQ as well as the amount of lead found in their bloodstream. Children are then broken into three groups: a group that has a low amount of lead in their blood, a group that has a medium amount of lead in their blood, and a group that has a high amount of lead in their blood. Statistics for these samples are listed in the table below:

Statistics Low Lead Group Medium Lead Group High Lead Group

Mean IQ 102.7 94.1 94.2

Standard Deviation 16.8 15.5 11.4

Sample Size 78 22 21

• Using statistical software, they obtain the following ANOVA output:

Df Sum of Squares Mean Square F Pr(>F)

Lead Level 2 2022.730 1011.365 4.071 0.01951

Residuals 118 29314.047 248.42

Questions

Problem 1

Before performing any further analysis, researchers wish to determine if there is a statistical difference in average IQ scores between some of the lead levels.

A. (2 points) Write the null and alternative hypotheses for a test to see if there is a difference in average IQ scores between some of the groups.

B. (2 points) Based on an examination of the box plots, do you believe there is a difference in the average IQ scores between some of the groups? Explain.

C. (2 points) At the 0.05 significance level, can we conclude that there is a difference in average IQ scores between some of the groups? Explain.

D. (2 points) Does the result of the hypothesis test change if we reduce the significance level to 0.01? Explain why or why not.

Problem 2

Researchers suspect that greater lead exposure results in lower IQ scores. Based on this, they claim that children in the "low lead" group will have a higher average IQ than children in the "high lead" group. They conduct a hypothesis test at the 5% significance level.

A. (2 points) State the null and alternative hypotheses.

B. (2 points) Are the conditions for inference satisfied? Explain.

C. (2 points) Compute the test statistic. For full credit, be sure to show your work.

D. (2 points) Determine the p-value or the critical value (depending on the method you prefer). Either way, be sure to show your work by drawing a sketch.

E. (2 points) Using either the p-value or critical value method, make a decision about the null hypothesis.

F. (1 point) Based on the results of the hypothesis test, is there convincing evidence that children in the "low lead" group have a higher average IQ than children in the "high lead" group?