. (8 pts) In each case, consider what you know about the distribution and then explain why you would expect it to be or not to be normally distributed.

a. The lifetimes of a set of batteries manufactured by a company in your town.

b. The list of guesses of the number of marbles in a large jar is filled with marbles at a neighborhood picnic.

c. The scores at an archery contest.

d. The heights of all the 1st grade students at your school

It would be interesting to read your thoughts about these different situations.

I would expact (a) and (b) to be normal distributions. (d) might be bimodal, with separate peaks for boys and girls, although heights tend to be about the same at that age. (c) might have a cluster of high scores with a rapid falloff at the high end and a slower falloff at the low scoring end of the distribution, it there are many poor archers entered.

a. The lifetimes of a set of batteries manufactured by a company in your town may not be normally distributed. This is because the lifetime of batteries can be affected by various factors such as quality of materials used, manufacturing process, and user behavior. It is likely that there will be a range of lifetimes, with some batteries lasting longer than others. This can lead to a distribution that is skewed, with a tail on one side indicating a longer lifetime for some batteries.

b. The list of guesses of the number of marbles in a large jar at a neighborhood picnic is expected to be normally distributed. When people submit their guesses, there is no specific distribution imposed on them and they are likely to make random guesses. According to the central limit theorem, when you have a large number of independent and identically distributed random variables (in this case, the guesses), the distribution of their average tends to be normal. Therefore, the distribution of guesses will likely resemble a normal distribution.

c. The scores at an archery contest might be normally distributed. Assuming that the archers are of similar skill levels and that there are no external factors that significantly affect their performance such as wind or equipment malfunction, the scores should follow a normal distribution. This is because many natural phenomena tend to follow a normal distribution, and the skill level of the archers can be considered a combination of various factors that may contribute to a normal distribution of scores.

d. The heights of all the 1st grade students at your school are not likely to be normally distributed. Human height is known to follow a roughly normal distribution in the general population, but it is also influenced by various genetic and environmental factors. In a specific group of 1st grade students, there could be variations in height due to factors such as age, gender, maturity, and growth patterns. These variations can result in a distribution that is not perfectly normal, with potential skewness or outliers in the data.