For each of these problems, solve it by hand using z-scores and the normal area table, then verify your answer with R.

1. The lifetimes of a certain brand of light bulb are normally distributed, with a mean of 3.5 years and a standard deviation of .75 years. What is the probability one of these light bulbs burns out in under two years?

2. Suppose the sizes of a group of giant squid are estimated to be normally distributed, with mean 37 feet, and standard deviation 3.5 feet. The largest recorded giant squid was around 43 feet long, but folklore refers to giant squid over 50 feet long. According to this data, what is the probability of seeing a giant squid of this size or larger?

3. The heights of adult men in the U.S. are normally distributed, with a mean of 70 inches and standard deviation of 3 inches. However, the average basketball player in the NBA ranges from 77 to 79 inches. Estimate the percentage of the population with heights in this range.

4. Grades on a standardized test are normally distributed with a mean of 82 and a standard deviation of 5. What would you have to score to place in the top ten percent of all test takers?

5. Strictly speaking, normal approximations to these data sets lead to some strange predictions. For instance, what is the probability of an adult man having a negative height based on the assumptions in problem 3? (Just use R here, the table won't help for such a small z-score) Why is this not a problem for the use of the normal approximation of data?