# Normal Probability & Area under the Standard Normal Curve

Included with each section or problem are reference examples and end of section exercises that can be used as a guide. Be sure to show your work in case partial credit is awarded. To receive full credit, work must be shown if applicable.

Section 5.1: Introduction to Normal Distribution and the Standard Normal Distribution

1. Use the Standard Normal Distribution table to find the indicated area under the standard normal curve.

(References: example 3 - 6 pages 244 - 247, end of section exercises 21 - 40 page 249)

a. Between z = 0 and z = 1.24

b. To the right of z = 1.09

c. Between z = -1.56 and z = -0.15

d. To the left of z = -1.93

Section 5.2: Normal Distributions: Find Probabilities

(References: example 1 and 2 page 253, end of section exercises 13 - 30 pages 257 - 259

2. The diameters of a wooden dowel produced by a new machine are normally distributed with a mean of 0.55 inches and a standard deviation of 0.01 inches. What percent of the dowels will have a diameter less than 0.57?

3. The a loan officer rates applicants for credit. Ratings are normally distributed. The mean is 240 and the standard deviation is 50. Find the probability that an applicant will have a rating less than 260.

Section 5.3: Normal Distributions: Finding Values

4. Answer the questions about the specified normal distribution. (References: example 4 and 5 page 264 - 265, end of section exercises 39 - 45 pages 267 - 268)

a. The lifetime of ZZZ batteries are normally distributed with a mean of 280 hours and a standard deviation  of 10 hours. Find the number of hours that represent the 25th percentile.

b. Scores on an English placement test are normally distributed with a mean of 50 and standard deviation  of 2.5. Find the score that marks the top 10%.

Section 5.4: Sampling Distribution and the Central Limit Theorem

5. Find the probabilities.

(References: example 5 and 6 page 276 - 277, end of section exercises 25 - 31 pages 280 - 281)

a. From National Weather Service records, the annual snowfall in the TopKick Mountains has a mean of 72 inches and a standard deviation  of 12 inches. If the snowfall from 25 randomly selected years are chosen, what it the probability that the snowfall would be greater than 75 inches?

b. The loan officer rates applicants for credit. Ratings are normally distributed. The mean is 240 and the standard deviation is 60. If 36 applicants are randomly chosen, what is the probability that they will have a rating less than 260?

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#### Solution Summary

The solution provides step by step method for the calculation of probability using the Z score and area under the standard normal curve. Formula for the calculation and Interpretations of the results are also included.

Normal Distribution, Standard Normal Distribution, Probability, Bell Curve and z-Scores

1. This week we learned about normal distributions:

a. Using your own words, tell me what the difference is between a normal distribution and a standard normal distribution.

b. Why do we convert values to z scores to find probability?

2. What is the difference between z scores and area in a bell curve?

3. T / F The number associated with the area under the bell curve that falls to the left of the mean or center line is a negative value (read this carefully).

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