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# Distributions, Standard Deviations and Z-Scores

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4. James Johnson, manager of quality control at Creative Auto Corp., just received a report from the assembly plant. The latest shipment of 200 lug bolts used to attach the wheels showed a mean diameter of 18.01 mm and a median of 17.92 mm. Therefore, James can conclude that the distribution of the diameters of the lug bolts
a. is perfectly symmetric.
b. is skewed left.
c. is skewed right.
d. has a range of 0.09 mm.

5. Juan Salvador just completed a study of the life expectancy of 100 light bulbs and discovered that the mean time that they lasted before burning out was 1,900 hours. If the standard deviation was 150 hours, the empirical rule allows Juan to conclude that approximately 68 of the bulbs burned out between _________ and _________ hours.
a. 1,750, 2,050
b. 1,800, 1,900.
c. 1,600, 2,200.
d. 1,450, 2,350.

Use the following information to answer questions 7 - 8:
Approximately 25% of the population belongs to a health maintenance organization (HMO). Assume that for a randomly selected group of 20 adults, the number belonging to an HMO has a binomial distribution.

7. The probability of finding exactly 5 in the 20 who belong to an HMO is:
a. 0.2023
b. 0.1567
c. 0.1750
d. 0.2345

8. The probability of finding at least one in the 20 who belongs to an HMO is:
a. 0.7500
b. 0.8012
c. 0.9056
d. 0.9968

9. The standard normal table shows an area value of 0.1 for a z-score of 0.25 and an area value of 0.35 for a z-score of 1.04. What percentage of the observations of a random variable that is normally distributed will fall between 0.25 standard deviations below the mean and 1.04 standard deviations above the mean?
a. 25%
b. 35%
c. 45%
d. 55%

10. Trudy Jones recently completed her certification examination and learned that her z-score was -2.5. The examining board also informed her that a failure to pass would be all scores that were 1 or more standard deviations below the mean and that those with scores higher than 2 standard deviations above the mean would receive a special commendation award. Trudy can, therefore, conclude that she
a. failed the exam.
b. passed the exam.
c. passed the exam and will receive a special commendation award.
d. passed the exam, but no commendation award is forthcoming.

https://brainmass.com/statistics/z-test/distributions-standard-deviations-and-z-scores-90329

#### Solution Summary

Distributions, Standard Deviations and Z-Scores are investigated. The solution is detailed and well presented.

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## Probabilities, z scores and distributions

1. Two hundred raffle tickets are sold. Your friend has 5 people in her family who each bought two raffle tickets. What is the probability that someone from her family will win the raffle?

a. What does it mean to say that x = 152 has a standard score of +1.5?
b. What does it mean to say that a particular value of x has a z-score of -2.1?
c. In general, what is the standard score a measure of?

3. Jolie has a time of 45 minutes for doing her statistics homework. If the mean is 38 minutes and the standard deviation is 3, calculate Jolie's z score. Once calculated, interpret your findings in terms of Jolie's performance (HINT: use the normal distribution and the probability that other students performed better or worse. You will want to think about your z score in terms of standard deviation units.)

4. How does the bell-shaped curve for the sampling distribution of sample means for samples of size n = 100 compare to the bell-shaped curve for the sampling distribution of sample means for samples of size n = 60? Think about it in terms of the standard error of the mean.

5. What are the characteristics of the normal distribution? Why is the normal distribution important in statistical analysis? Provide an example of an application of the normal distribution.

6. In your own words describe the standard normal distribution. Explain why it can be used to find probabilities for all normal distributions.

7. Data is collected from a large Midwestern city. The growth of these adolescents over the course of puberty is normally distributed with a mean of 5.26 inches and a standard deviation of 0.50 inches.

a. What percentage of the adolescents in this city grew less than 4.5 inches? (HINT: convert your score of interest to a standard score and draw a graph to represent the information you want to obtain. Use your table to calculate proportions which are easily converted to percentages by multiplying by 100).

b. What percentage of the adolescents in this city grew more than 5.12 inches?

c. A random sample of 100 adolescents is gathered and the mean growth during puberty was 5.12. If another sample of 100 is taken, what is the probability that its sample mean will be greater than 5.12 inches? (HINT: We are now doing inferential statistics. Identify your population mean, your sample mean, the population standard deviation, and N which is sample size)

d. Why is the z-score used in answering (a), (b), and (c)?
e. Why is the formula for z used in (c) different from that used in (a) and (b)?

8. Assume that the population of heights of male college students is approximately normally distributed with mean m of 68 inches and standard deviation s of 3.75 inches. A random sample of 16 heights is obtained.

a. Describe the distribution of x, height of the college student.
b. Find the proportion of male college students whose height is greater than 70 inches.
c. Describe the distribution of , the mean of samples of size 16.
d. Find the mean and standard error of the distribution.
e. Find P ( > 70
f. Find P ( < 67)

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