# 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?

2. Answer the following:

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)

https://brainmass.com/statistics/probability/probabilities-z-scores-and-distributions-271754

#### Solution Summary

The solution provides step by step method for the calculation of probability using the Z score. Formula for the calculation and Interpretations of the results are also included. The solution also provides characteristics, importance and application of normal distribution and a short description on Standard normal distribution.

Probability Distribution, Binomial, Random Samples, & z-scores

1. Determine whether each of the distributions given below represents a probability distribution. Justify your answer.

(A) x 1 2 3 4

P(x) 1/12 5/12 1/3 1/12

(B)x 3 6 8

P(x) 2/10 .5 1/5

(C)x 20 35 40 50

P(x) 0.4 -0.2 0.5 0.3

2. Consider a binomial distribution with 14 identical trials and a probability of success of 0.4.

i. Find the probability that x = 3 using the binomial tables.

ii. Use the normal approximation to find the probability that x = 3. Show all work.

3. The diameters of oranges in a certain orchard are normally distributed with a mean of 4.85 inches and a standard deviation of 0.40 inches. Show all work.

(A) What percentage of the oranges in this orchard have diameters less than 6.3 inches?

(B) What percentage of the oranges in this orchard are larger than 4.75 inches?

(C) A random sample of 100 oranges is gathered and the mean diameter obtained was 4.75. If another sample of 100 is taken, what is the probability that its sample mean will be greater than 4.75 inches?

(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)?

4. Assume that the population of heights of male college students is approximately normally distributed with mean u of 68 inches and standard deviation sigma of 3.75 inches. A random sample of 16 heights is obtained. Show all work.

(A) Find the proportion of male college students whose height is greater than 70 inches.

(B) Find the mean and standard error of the distribution.

(C) Find P( > 70)

5. Answer the following questions regarding the normal, standard normal and binomial distributions.

(A) What conditions must be met in order to use the normal distribution to approximate the binomial distribution?

(B) How does the standard normal distribution differ from the normal distribution?

(C) Why is the correction for continuity necessary when the normal distribution is used to approximate a binomial distribution?

6. Four green marbles are selected, one at a time from a bin of marbles containing 6 black, 6 red and 6 green marbles. Let x represent the number of green marbles drawn in 4 draws.

(A) If this experiment is completed without replacing the marbles, explain why x is not a binomial random variable.

(B) If this experiment is completed with replacement of the marbles, explain why x is a binomial random variable.