Normal Curves

Normal Curves AP Statistics Name: ______Date:______Period: ______

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. Normal Curves AP Statistics 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? Normal Curves AP Statistics

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 < 2.85 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 µ=0.37 and standard deviation σ=0.04. For each part that follows, sketch and shade an appropriate Normal distribution. Then show your work. a) What proportion of adhesions measured in use are higher than 0.40?

(a) We want to find the area under the N(0.37, 0.04) distribution to the right of 0.4. The graphs below show that this area is equivalent to the area under the N(0, 1) distribution to the right of z=(0.4− 0.37)/.04=0.75.

Using Table A, the proportion of adhesions higher than 0.40 is 1 − 0.7734 = 0.2266. b) What proportion of adhesions are between 0.40 and 0.50?

We want to find the area under the N(0.37, 0.04) distribution between 0.4 and 0.5. This area is equivalent Normal Curves AP Statistics to the area under the N(0, 1) distribution between z = = 0.75 and 0.04 z = = 3.25 . (Note: New graphs are not shown, because they are almost identical to the graphs above. The shaded region should end at 0.5 for the graph on the left and 3.25 for the graph on the right.) Using Table A, the proportion of adhesions between 0.4 and 0.5 is 0.9994 − 0.7734 = 0.2260. c) Improvements in the locomotives computer controls change the distribution of adhesion to a Normal distribution with mean µ = 0.41 and standard deviation σ=0.02. Find the proportion in a) and b) after this improvement.

Now, we want to find the area under the N(0.41, 0.02) distribution to the right of 0.4. The graphs below show that this area is equivalent to the area under the N(0, 1) distribution to the right of z = = − 0.5 .

Using Table A, the proportion of adhesions higher than 0.40 is 1 − 0.3085 = 0.6915. The area under the N(0.41,

0.02) distribution between 0.4 and 0.5 is equivalent to the area under the N(0,1) distribution between z = = − 0.5 and z = = 4.5 . Using Table A, proportion of adhesions between 0.4 and 0.5 is 1 − 0.3085 = 0.6915. The proportions are the same because the upper end of the interval is so far out in the right tail.

16. Use Table A to find the value z from a standard distribution that satisfies each of the following conditions. (Use the value of z from Table A that comes closest to satisfying the condition.) In each case, sketch a standard Normal curve with your value of z marked on the axis. a) The point z with 25% of the observations falling to the left of it.

(a) The closest value in Table A is −0.67. The 25th percentile of the N(0, 1) distribution is −0.67449. Normal Curves AP Statistics b) The point z with 40% of the observations falling to the right of it.

(b) The closest value in Table A is 0.25. The 60th percentile of the N(0, 1), distribution is 0.253347.

17. Scores on the Wechsler Adult Intelligence Scale (a standard IQ test) for the 20 to 34 age group are approximately Normally distributed with µ = 110 and σ = 25. For each part that follows, sketch and shade an appropriate Normal distribution. Then show your work. a) What percent of people aged 20 to 34 have IQ scores above 100?

b) What percent have scores above 150?

c) MENSA is an elite organization that admits as members people who score in the top 2% on IQ tests. What score on the Wechsler Adult Intelligence Scale would an individual have to earn to qualify for MENSA membership? Normal Curves AP Statistics