# Solutions Homework 2

**Topics:**Normal distribution, Standard deviation, Intelligence quotient

**Pages:**7 (1266 words)

**Published:**January 22, 2015

1. Where on the normal curve are the inflection points located? Where the slope starts to drop off at.

2. What is the standard normal distribution?

A standard Normal distribution is a Normal curve with a mean of 0 and a standard deviation of 1.

3. What information does the standard normal table give?

The area that falls to the left of the given z-score.

4. How do you use the standard normal table (Table A) to find the area under the standard normal curve to the left of a given z-value? Draw a sketch. Locate the z-score to the nearest 10th along the left hand side, then follow it to the right to find the hundredth and look at the corresponding proportion. 5. How do you use Table A to find the area under the standard normal curve to the right of a given z-value? Draw a sketch. Subtract the proportion you find from 1.

6. How do you use Table A to find the area under the standard normal curve between two given z-values? Draw a sketch. Find the z-scores for both given values of x. Find the proportions in Table A. Subtract the smaller proportion from the larger on. 10. Below are two normal curves, both with mean 0. Approximately what is the standard deviation of each curve? For the narrow one, about 0.3. For the shorter one, about 0.6.

11. The distribution of heights of adult American men is approximately Normal with mean 69 inches and standard deviation 2.5 inches. Draw a Normal curve on which this mean and standard deviation are correctly located. (Hint: Draw the curve first, locate the points where the curvature changes, then mark the horizontal axis.)

12. The distribution of heights of adult American men is approximately Normal with mean 69 inches and standard deviation 2.5 inches. Use the 68-95-99.7 rule to answer the following questions. a) What percent of men are taller than 74 inches?

b) Between what heights do the middle 95% of men fall?

c) What percent of men are shorter than 66.5 inches?

d) A height of 71.5 inches corresponds to what percentile of adult male American heights? (a) Approximately 2.5% of men are taller than 74 inches, which is 2 standard deviations above the mean. (b) Approximately 95% of men have heights between 69−5=64 inches and 69+5=74 inches. (c) Approximately 16% of men are shorter than 66.5 inches, because 66.5 is one standard deviation below the mean. (d) The value 71.5 is one standard deviation above the mean. Thus, the area to the left of 71.5 is the 0.68 + 0.16 = 0.84. In other words, 71.5 is the 84th percentile of adult male American heights. 13. Each year the school buys flares to light the J hill for homecoming. The flares are designed to last 30 minutes total. The time the flares actually last is approximately Normally distributed with a mean time of 30 minutes and a standard deviation of 2 minutes. a) Draw an accurate sketch of the distribution of total time burning for flares lit for the J. Be sure to label the mean, as well the points one, two, and three standard deviations away from the mean on the horizontal axis.

b) A flare that lasts for 32.5 minutes is at what percentile in this distribution?

c) What percent of flares will last longer than 35 minutes?

14. Use Table A to find the proportion of observations from a standard Normal distribution that satisfies each of the following statements. In each case, sketch a standard Normal curve and shade the area under the curve that is the answer to the question. a) z <

b) z > 2.85

c) z > -1.66

d) -1.66 < z < -1.27

(a) 0.9978 (b) 1 − 0.9978 = 0.0022 (c) 1 – 0.0485 = 0.9515 (d) 0.9978 – 0.0485 = 0.9493

15. An important measure of the performance of a locomotive is its “adhesion,” which is the locomotive’s pulling force as a multiple of its weight. The adhesion of one 4400-horsepower diesel locomotive varies in actual use according to a Normal distribution with mean...

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