Information from the American Institute of Insurance indicates the mean amount o
ID: 2955923 • Letter: I
Question
Information from the American Institute of Insurance indicates the mean amount of lifeinsurance per household in the United States is $110,000. This distribution follows the
normal distribution with a standard deviation of $40,000.
a. If we select a random sample of 50 households, what is the standard error of the mean?
b. What is the expected shape of the distribution of the sample mean?
c. What is the likelihood of selecting a sample with a mean of at least $112,000?
d. What is the likelihood of selecting a sample with a mean of more than $100,000?
e. Find the likelihood of selecting a sample with a mean of more than $100,000 but less
than $112,000.
Explanation / Answer
Shape of a distribution refers to the mathematical formula that describes the distribution of scores. One measure is central tendency which can be a mean (average) mode (most common score) median( score above and below which 50% of the scores lie. Each of these have different math properties. For describling the normal distribution the mean is the most common. It just happesn that inthe normal distribution the mean=mode=median. Thsi not rue in all distributions. the Normal Distribution is a SPECIAL distribution. Dispersion is another measure of a distrbuitons of scores. It refers to how scattered the scores are or how broad the range od scores are. RANGE and variance and standard vaiance are measures of dispersion. Examples of dispersion: 3,3,3 dispersion = 0 they are all same. mean =9 2,3,4 disperaiosn higher mean = 9 -1,3,8 highest disperion with mena = 9 The normal disribtion is well known mathematically with mean = 0 and std. deviaion of 1. this is the N(0,1) disribution. The N(0,1) is called a bell shape cure because if you graphed thousands of points the outline would look like the shape of the bell. You expect most of the scores to be in the center (mean) and very few really low scores and very few very high scores. Most people are average IQ (100) very few are low 60 or very high 200. Or whink of weight: most people are normal; a few are obese and some are anorexic. Your AII curve would be: N(110,000, 40,000) Since the AII curve is noramlly shaped you would expect that when SAMPLING scores and calculating a MEAN you would get the disribtuion of these means to look like the original curve - noraml shapes bell curve. Note the mean and variance measures may change but with a LARGE number of samples taken the shape will approach the bell shape. -------------- A distribution that is not bell shaped is SKEWED left or right. If IQ mean is 100 but the kids in your school average 150 with a few low scores, the curve has a bulge at the right end and a long skinny tail on the left. Note that the median, mean, and mode will not be the same value. Variance will not be 1 that ells you it is skewed and not a noraml distribution.