# Probabilities using the Normal Distribution

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1. True or False? A normal distribution is a continuous probability distribution used with continuous random variables.

2. If a z-score is close to −3.49, the cumulative area is close to what value?

3. What is the area under a standard normal curve between z = −1.47 and z = 1.58?

4. What is the area under a standard normal curve to the right of z = 0.35?

5. What is the area under a standard normal curve between z = −.80 and z = 1.35?

6. The normal distribution is a type of probability density function. Areas found under this curve are equivalent to probabilities. For example, the probability that z lies between a and b under the standard normal distribution is denoted P(a < z < b) and this probability is the same as the area under the curve between the z-scores, z1 = a and z2 = b. Using a standard normal curve, what is P(1.12 < z < 1.90)?

7. U.S. Women Ages 55-64: Total Cholesterol Suppose the total cholesterol in women follows a normal distribution. Suppose µ = 219 and σ = 41.6. What is P(200 < x < 239)?

Problems 8-10. Use the Information below to answer the questions.

Fish Lengths The lengths of Atlantic croaker fish are normally distributed, with a mean of 10 inches and a standard deviation of 2 inches. An Atlantic croaker fish is randomly selected.

8. Find the probability that the length of the fish is less than 7 inches.

9. Find the probability that the length of the fish is between 7 and 15 inches.

10. Find the probability that the length of the fish is more than 15 inches.

Problems 11 and 12. Use the Standard Normal Table to find the z-score that corresponds to the given cumulative area. If the area is not in the table, use the entry closest to the area. If the area is halfway between two entries, use the z-score halfway between the corresponding z-scores.

11. Area = 0.7324

12. Area = 0.1788

13. Find the z-score that has 54.7% of the distribution's area to its right.

Problems 17 and 18. Assume there is a sample with n = 145, µ = 29 and σ = 1.7.

17. What is P(x > 28.7)?

18. Would it be considered unusual if x > 28.7?

19. Gas prices. During a certain week the mean price of gasoline was $3.305 per gallon. A random sample of 38 gas stations is drawn from this population. What is the probability that the mean price for the sample was between $3.310 and $3.320 that week? Assume σ = $0.049.

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This solution consists of lots of problems of finding probabilities using normal distributions.

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1. True or False? A normal distribution is a continuous probability distribution used with continuous random variables.

True

2. If a z-score is close to −3.49, the cumulative area is close to what value?

0.0002

3. What is the area under a standard normal curve between z = −1.47 and z = 1.58?

0.8722

4. What is the area under a standard normal curve to the right of z = 0.35?

0.3632

5. What is the area under a standard normal curve between z = −.80 and z = 1.35?

0.6996

6. The normal distribution is a type of probability density function. Areas found under this curve are equivalent to probabilities. For example, the probability that z lies between a and b under the standard normal distribution is denoted P(a < z < b) and this probability is the same as the area under the curve between the z-scores, z1 = a and z2 = b. Using a standard normal curve, what ...

###### Education

- BSc , Wuhan Univ. China
- MA, Shandong Univ.

###### Recent Feedback

- "Your solution, looks excellent. I recognize things from previous chapters. I have seen the standard deviation formula you used to get 5.154. I do understand the Central Limit Theorem needs the sample size (n) to be greater than 30, we have 100. I do understand the sample mean(s) of the population will follow a normal distribution, and that CLT states the sample mean of population is the population (mean), we have 143.74. But when and WHY do we use the standard deviation formula where you got 5.154. WHEN & Why use standard deviation of the sample mean. I don't understand, why don't we simply use the "100" I understand that standard deviation is the square root of variance. I do understand that the variance is the square of the differences of each sample data value minus the mean. But somehow, why not use 100, why use standard deviation of sample mean? Please help explain."
- "excellent work"
- "Thank you so much for all of your help!!! I will be posting another assignment. Please let me know (once posted), if the credits I'm offering is enough or you ! Thanks again!"
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