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# Basic Statictcal problems

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A local bottling company has determined the number of machine breakdowns per month and their respective probabilities as shown below:

Number of
Breakdowns Probability
0 0.12
1 0.38
2 0.25
3 0.18
4 0.07

____ 1. Refer to Exhibit 1-1. The expected number of machine breakdowns per month is
a. 2
b. 1.70
c. one, since it has the highest probability
d. at least 4

____ 2. During the last 10 years marketing executives believed that the same proportion of adult men and adult
women watched TV news programs. Current information indicates there has been an increase in the
percentage of women watching TV news programs. The marketing executives would like to perform a
statistical test to assure themselves that a greater proportion of women now watch TV news programs than
men. What should be used for the Null and Alternative hypotheses to perform this test?

Exhibit 1.2.
Consider the letters involved in the word STATISTICS.
____ 3. Refer to Exhibit1.2 above. How many "bars" would be involved if you were supposed to construct a bar
graph to represent the frequency distribution of letters?
a) 6
b) 10
c) 8
d) 5

____ 4 Refer to Exhibit 1.2 above. What is the relative frequency of letter I?
a) 2
b) 0.2
c) 0.02
d) 0.4

____ 5. Twenty percent of the students in a class of 100 are planning to go to graduate school. The standard
deviation of this binomial distribution is
a. 20
b. 16
c. 4
d. 2
____ 6. Assume that you have a binomial experiment with p = 0.5 and a sample size of 100. The expected value of
this distribution is
a. 0.50
b. 0.30
c. 100
d. 50
_____ 7. A small company has ten employees with a mean annual salary of \$75,000. The company is planning to
expand its activities by hiring an administrative assistant and a consultant with annual salaries of \$45,000 and \$95,000,
respectively. What will the mean annual employee salary be if the company expands its workforce as planned?
a) \$17,916.67 b) \$71,666.67 c) \$74,166.67 d) \$21,500

Problem
_____ 8. The following frequency distribution shows the yearly tuitions (in \$1,000s) of a sample of private colleges.

Yearly
Tuition Frequency
12 - 16 5
17 - 21 4
22 - 26 3
27 - 31 2

The mean yearly tuition is:
a) \$19,714.29 b) \$1,971.40 c) \$2000.00 b) \$2,000.00

Exhibit 1-3
AMR is a computer consulting firm. The number of new clients that they have obtained each month has
ranged from 0 to 6. The number of new clients has the probability distribution that is shown below.

Number of
New Clients Probability
0 0.05
1 0.10
2 0.15
3 0.35
4 0.20
5 0.10
6 0.05

____ 9. Refer to Exhibit 1-3. The variance is
a. 1.431
b. 2.047
c. 3.05
d. 21
____ 10. If a population data set is normally distributed, what is the proportion of measurements you would expect to fall
within &#956; ± &#963;?
a) 100% b) 95% c) 50% d) 68%

____ 11. A bottling company needs to produce bottles that will hold 8 ounces of liquid for a local brewery. Periodically,
the company gets complaints that their bottles are not holding enough liquid. To test this claim, the bottling
company randomly samples 64 bottles and finds the average amount of liquid held by the 64 bottles is
7.9145 ounces with a standard deviation of 0.40 ounce. Suppose the p-value of this test turned out to be 0.0436. State the proper conclusion.
a) At &#945; = 0.035, accept the null hypothesis.
b) At &#945; = 0.05, reject the null hypothesis.
c) At &#945; = 0.085, fail to reject the null hypothesis.
d) At &#945; = 0.025, reject the null hypothesis.

Exhibit 1-4
The weight of items produced by a machine is normally distributed with a mean of 8 ounces and a standard deviation of 2 ounces.
____ 12. Refer to Exhibit 1-4 above. What percentage of items will weigh at least 11.7 ounces?
a. 46.78%
b. 96.78%
c. 3.22%
d. 53.22%

___ 13. For a standard normal distribution, the probability of obtaining a z value between -2.4 to -2.0 is
a. 0.4000
b. 0.0146
c. 0.0400
d. 0.5000

Exhibit 1-5
y = 28.886x + 296.52
R2 = 0.3244
0
500
1000
1500
2000
20 25 30 35 40 45 50
Newspaper Ads in Thousands of Dollars
Sales in Thousands of
Dollars

____ 14. Refer to Exhibit 1.5 above, use the above line to predict the sales if the amount of newspaper ads is
\$42,000.
a) \$1,418,321
b) \$1,509,732
c) \$1,616,789
d) \$1,943,345

____ 15. Suppose Stat I students' ages follow a skewed right distribution with a mean of 24 years old and a standard
deviation of 2 years old. If we randomly sampled 150 students, which of the following statements about the
sampling distribution of the sample mean age is incorrect?
a) The mean of the sampling distribution is approximately 24 years old.
b) The standard deviation of the sampling distribution is equal to 2 years old.
c) The shape of the sampling distribution is approximately normal.
d) None of the above statements are correct.

Exhibit 1.6
HYPOTHESIS: VARIANCE X = x
X = gpa
SAMPLE MEAN OF X = 3.2854
SAMPLE VARIANCE OF X = .22000
SAMPLE SIZE OF X = 243
HYPOTHESIZED VALUE (x) = 3.4
VARIANCE X - x = -.1146
z = -3.80870
____ 16 Find za/2 for &#945; = 0.01.
a) 2.33
b) 1.645
c) 1.96
d) 2.575

Exhibit 1.7.
A statistical experiment involves flipping a coin that is much thicker than a regular coin of the same diameter.
When flipped, the probability for the thick coin to come to rest on its edge is 4 percent; otherwise, it is a "fair coin" in the sense that we can get heads or tails with equal probabilities.

____ 17 Refer to Exhibit 1.7 above. What is the total number of possible outcomes comprising the sample space for
the statistical experiment?
a) 1 b) 2 c) 3 d) 4

____ 18 Refer to Exhibit 1.7 above. What is the probability of getting heads when conducting the statistical experiment?
a) 0.54 b) 0.33 c) 0.46 d) 0.48

____ 19. Given that Z is a standard normal random variable, what is the value of Z if the area to the left of Z is 0.9382?
a. 1.8
b. 1.54
c. 2.1
d. 1.77

_ __ 20. High temperatures in a certain city for the month of August follow a uniform distribution over the interval 145o F
to 165o F What is the probability that a randomly selected August day has a high temperature that exceeded
150o F?
a) .75.
b) .25
c) .05
d) .025

____ 21. If a hypothesis test were conducted using &#945; = 0.10, for which of the following p-values would the null
hypothesis be rejected.
a) 0.15 b) 0.110 c) 0.090 d) 0.105

____ 22. Suppose you are interested in conducting the statistical test of H0: u = 255 against Ha : u > 255, and you have
decided to use the following decision rule: Reject H0 if the sample mean of a random sample of 81 items is more than 270. Assume that the standard deviation of the population is 63.
Express the decision rule in terms of z.
a) z = 2.14 b) z < 2.14 c) z > 2.14 d) z &#8800; 2.14

Solve the Problem
____ 23. Farmers often sell fruits and vegetables at roadside stands during the summer. One such roadside stand has
a daily demand for tomatoes that is approximately normally distributed with a mean equal to 135 tomatoes per day and a
standard deviation equal to 30 tomatoes per day. How many tomatoes must be available on any given day so that there
is only a 1.5% chance that all tomatoes will be sold?
a) 200 b) 187 c) 160 d) 130

____ 24. Given H0: &#956; = 25, Ha: &#956; &#8800; 25, and P = 0.041. Do you reject or fail to reject H0 at the 0.01 level of significance?
a) not sufficient information to decide
b) reject H0
c) fail to reject H0
d) fail to accept Ha

_____ 25. A bakery has determined that the number of loaves of its white bread demanded daily has a normal
distribution with a mean 7200 loaves and a standard deviation of 300 loaves. Based on cost considerations,the company has decided that its best strategy is to produce a sufficient number of loaves so that it will be fully supplied on 94% of all days. How many loaves of bread should the company produce?
a) 4678 b) 7667 c) 6800 d) 7130

_____ 26. Referring to Question #29, onwhat percentage of days will the company be left with more than 500 loaves of
a) 42.67% b) 45.62% c) 54% d) 12%

TRUE/FALSE. Write 'T' if the statement is true and 'F' if the statement is false.
Answer the question True or False.
____ 27. If A and B are independent events, then A and B are mutually exclusive also.
____ 28. A Type I error occurs when we accept a false null hypothesis.
____ 29. The binomial distribution can be used to model the number of rare events that occur over a given time period.
____ 30. One drawback of pie charts, dot plots, stem-and-leaf displays and histograms is that no measure of reliability can be attached to a graph.