MATH 221 Week 5 DQ Interpreting Normal Distributions

This file of MATH 221 Week 5 DQ Interpreting Normal Distributions comprises:

Assume that a population is normally distributed with a mean of 100 and a standard deviation of 15. Would it be unusual for the mean of a sample of 3 to be 115 or more? Why or why not?

PLEASE COMPLETE THE QUIZ THIS WEEK based on Week 3 and Week 4 materials only.

Here are a few questions that you can participate for discussion points from your study plan on normal distribution that may help you toward understanding concepts and the quiz for Week 7. Please work on only one question at a time. Answers are posted at the end.

OBJECTIVE: CONCEPTS

1. What requirements are necessary for a normal probability distribution to be a standard normal distribution?

STUDY PLAN: 5.1.2, 5.1.3, 5.1.7

OBJECTIVE: COMPUTE AND INTERPRET Z-SCORES of NORMAL DISTRIBUTIONS

2. The systolic blood pressures of a sample of adults are normally distributed, with a mean pressure of 115 millimeters of mercury and a standard deviation of 3.6 millimeters of mercury. The systolic blood pressures of four adults selected at random are 122, 113, 106 and 128 millimeters of mercury. The graph of the standard normal distributions is shown below. Complete a) and b) PLEASE SEE ATTACHED DOCUMENT FOR GRAPHS

a. Without converting to z scores, match the values with the letters A, B, C, and D on the given graph of the standard normal distribution.

b. Find the z-score that corresponds to each value and check your answers to part (a) (Round to two decimals as needed)

STUDY PLAN: 5.1.41, 5.1.43

OBJECTIVE: FIND PROBABILITIES USING THE STANDARD NORMAL DISTRIBUTION

3. For the standard normal distribution shown below, find the probability of z occurring in the indicated on the graph. Please see attached document

STUDY PLAN: 5.1.45, 5.1.47, 5.1.55, 5.1.57

OBJECTIVE: FIND PROBABILITIES FOR NORMALLY DISTRIBUTED VARIABLES

4. Assume the random variable x is normally distributed with mean mu= 50 and

standard deviation sigma= 7. Find P(x > 42)

STUDY PLAN: 5.2.1, 5.2.3, 5.2.5, 5.2.7, 5.2.9, 5.2.11, 5.2.15

OBJECTIVE: APPLICATIONS OF NORMAL DISTRIBUTION

5. Use the normal distribution of SAT writing scores with mean = 493 and standard deviation = 111.

a. What percentage of SAT writing scores are less than 600?

b. If 1000 SAT writing scores are randomly selected, about how many would you expect to be greater than 550?

STUDY PLAN: 5.2.21, 5.2.23, 5.2.25, 5.2.27

OBJECTIVE: FIND A Z SCORE GIVEN THE AREA UNDER THE NORMAL CURVE

6. Find the z score that corresponds to the cumulative area of 0.049

STUDY PLAN: 5.3.1, 5.3.3, 5.3.5, 5.3.7, 5.3.17, 5.3.19, 5.3.21, 5.3.23, 5.3.25, 5.3.27, 5.3.29

OBJECTIVE: APPLICATION OF NORMAL DISTRIBUTION

7. In a survey of women in a certain country (ages 20- 29), the mean height was 65.3 inches with a standard deviation of 2.67 inches.

a. What height represents the 98th percentile?

b. What height represents the first quartile?

STUDY PLAN: 5.3.31, 5.3.33, 5.3.35, 5.3.38, 5.3.39, 5.3.41

OBJECTIVE: Interpret sampling distributions

8. A population has a mean mu= 86 and a standard deviation sigma =20. Find the mean and standard deviation of a sampling distribution of sample means with a sample size n= 268

STUDY PLAN: 5.4.1, 5.4.3

OBJECTIVE: CENTRAL LIMIT THEOREM

9. Use the central limit theorem to find the mean and standard error of the mean of the sampling distribution.

The mean price of photo printers on a website is $221 with a standard deviation of $69. Random samples of size 34 are drawn from the population and the mean of each sample is determined. (ROUND ANSWERS TO THREE DECIMALS)

10. The population mean annual salary for environmental compliance specialists is about $61,500. A random sample of 34 specialists is drawn from this population. What is the probability that the mean salary is less than $59,000? Assume standard deviation sigma= $5800?