# Central Limit Theorem and Probabilities Under Standard Normal Curve

Will you check my work on 1, 2 & 5 and then help me with 3, 4 & 6... I don't know if the formula is still the same for all of them or different.

1. The heights of eighteen-year-old men are approximately normally distributed with a mean (µ) of 68 inches and a standard deviation (σ) of 3 inches. What is the probability that an eighteen-year-old man selected at random is taller than 70 inches?

Z= 70-68 0.6667

3

2. Suppose that, instead of randomly selecting one 18-year-old male, you randomly select ten 18-year-old males. What is the probability that the average height for these 18-year-old males is more than 70 inches?

Z= 70-68 2.1 = 2.1 0.4821

3/√10

3. The average age for employees at an amusement park is 24 years old with a standard deviation of 2.5 years. Suppose random samples of 40 employees are selected. What would the distribution of average ages from samples of this size look like? Why?

4. What would the average be for all sample means from samples of this size?

5. What would the standard error be for all sample means from samples of this size?

2.5/√40 = 0.40

6. What is the probability if you randomly selected 40 employees and averaged their ages together that the sample mean would be between 23 and 25 years?

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#### Solution Preview

1. The heights of eighteen-year-old men are approximately normally distributed with a mean (µ) of 68 inches and a standard deviation (σ) of 3 inches. What is the probability that an eighteen-year-old man selected at random is taller than 70 inches?

Z=(70-68)/[3]=0.6667. Probability=P(Z>0.67)=0.2514

2. Suppose that, instead of randomly selecting one 18-year-old male, you randomly ...

#### Solution Summary

The solution gives detailed steps on computing probabilities either under standard normal curve or applying the central limit theorem.

Statistics: Normal Distribution, probabilities, values, central limit theorem

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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