The LEGO Group, an international company, makes the LEGO blocks that many of us have played with as kids. Many of its products require that the production process performs according to specifications. One of the products is Little People. The neck diameter of each of the Little People must be 0.5 inches so that it can be attached to the head properly. LEGO is interested in testing to see whether this process is performing according to specifications. If it is not functioning correctly, the LEGO Group will shut down the manufacturing process to fix the Little People machine.

1. The line manager takes a random sample of 40 Little People heads. The average diameter of the sampled necks is 0.48 inches, with a standard deviation of 0.05 inches. Is there sufficient evidence that the machine is not working correctly? Use a two-tailed hypothesis test with a significance level of ?=0.02 to determine the answer.
2. Give the confidence interval estimate for the actual population neck size of Little People. Use a 98% confidence level so that it aligns with the hypothesis test in part 2.
3. Does "0.5 inches" appear in the confidence interval? Does the confidence interval estimate of the mean neck size, of Little People align with the conclusion of your hypothesis test? Explain why the two methods for estimating match or don't.