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Fall 2007
Math 227
Final
Name: _____________________________________
You may use TI calculator for this test. However, you must show all details for hypothesis testing. For
confidence interval, you must show the critical value and the margin of error. Total Point: 165.
1.
(9 points, 3 each) Twelve
marbles are placed in a hat, three are black, two blue, one green, four white
and two red. Two marbles are drawn out at random, without replacement. Find the probability
that
(a). Both marbles are black.
(b).The first is red, and the second is green.
(c). Neither marble is white.
2.
(Total: 10 points) Slick
Motor Tire Inc. uses a tire mold which occasionally, and randomly, produces out
of-round tires. In fact 20% of the tires which it produces are out-of-round. Suppose Slick takes a
random sample of 10 tires produced by this tire mold.
(a). ( 5 pts) Find the expected number of out-of-round tires among these 10.
(b). ( 5 pts) Find the probability that exactly two of these 10 tires are out-of-round.
3.
(Total: 20 points) Consider
the following set of sample data, and find the following (note: if you are
using TI 83 to find these values, you need to write down the formulas that you should have used)
19
12
9
19
0
4
3
14
5
5
24
2
5
7
17
(a). (1 pt) Mean
(j). (4 pts ) Construct a frequency distribution Table.
Using 0 as the lower class limit of the first class and 5
as the class width.
(b). (1 pt) Median
(c). (1 pt) Mode
(d).
(1 pt) Standard
Deviation
(k).
(3 pts ) Construct
(l).
(2 pts ) Determine
a histogram. (use part (j))
(e). (1 pt) Variance
(f).
(2 pts) Find
the z-score for 14.
(i). (4 pts) Construct a Box-whisker plot for the data.
if the data is skewed left, right or
symmetric. (Use part (k))
4.
(Total: 15 points) Ted
regularly plays a target shooting competition in his yard. He is not very good at
the game and his score has the following probability distribution.
Score
0
1
2
3
Probability
5.
0.1
0.4
0.3
0.2
a.
( 5 pts)
Find the expected value and standard deviation of Ted’s score.
b.
( 5 pts)
Find the probability that Ted’s score is at most 2.
c.
( 5 pts)
Suppose he plays the game twice. Find the probability that his total is at least 5.
(Total: 10 points) The
speeds (in mph) of cars on Garfield Avenue are observed to be normally
distributed with mean 37 and standard deviation 4.5. The speed limit is 35 mph.
a. ( 5 pts) What fraction of drivers is exceeding the speed limit?
b.
Officer Jordan decides to give tickets only to the fastest 15% of cars on Garfield
Avenue. How fast must a driver go to get a ticket from Officer Jordan?
( 5 pts)
6.
(Total: 10 points) You
are constructing a marketing research study to determine what proportion of
consumers between the ages of 18 and 35 own DVD players. You have randomly sample 60
individuals in that age range, and you have determined that 11 of them own DVD players.
a. ( 5 pts) Find the 90% confidence interval for the proportion.
b.
7.
Suppose that your contract for the study requires that you estimate the proportion
within plus or minus 0.05 with 90% confidence. How many additional subjects would you
need to sample to obtain that level of precision?
( 5 pts)
(Total: 10 points) The
weights of packages carried by UPS courier service has a normal distribution with
a mean of 17.2 ounces and a standard deviation of 6.3 ounces.
a. ( 5 pts) It is proposed that packages weighing over 24 ounces will be subjected to a
surcharge. What proportion of packages will be affected by the surcharge?
b.
A random sample of 20 packages is taken. Find the probability that their average
weight is between 15 and 18 ounces.
( 5 pts)
8.
9.
(Total: 15 points) The
Monterey Park city council contracted a statistical consultant to prepare a study
about the impact of abolishing rent control regulations. As a preliminary step, the researcher
decided to estimate the mean rent increase for a one-bedroom apartment in Monterey Park. From
a simple random sample of 5 one-bedroom apartments in the area he obtained the following
monthly rent increases (in dollars): 130, 220, 175, 140, 100.
a. ( 4 pts) Find point estimates for the mean and standard deviation of the rent increase for a
one-bedroom apartment in Monterey Park.
b.
( 6 pts)
Assume a normal distribution for rent increases. Determine a 95% confidence
interval for the mean rent increase for one-bedroom apartments in Monterey Park.
c.
( 5 pts)
After the council meeting the consultant was asked to prepare a more detailed study
estimating the mean apartment rent increase within $10 with confidence of 99%. How
many apartments does he need to select in total (including the original 5) to obtain a
confidence interval with the required width?
Suppose that replacement times for washing machines are normally distributed with a
mean of 8.7 years and a standard deviation of 1.6 years. Find the replacement time that separates
the top 18% from the bottom 82%.
(6 points)
10. (Total: 20 points) The data in the table below were obtained through a survey of randomly selected
subjects.
Ticketed for Speeding Within the Last Year?
Yes
No
Men
26
224
Women
27
473
a.
( 4 pts)
If one of the survey subjects is randomly selected, find the probability of getting
someone ticketed for speeding.
b.
( 4 pts)
c.
( 4 pts)
d.
( 8 pts)
If one of the survey subjects is randomly selected, find the probability of getting a man
or someone ticketed for speeding.
Find the probability of getting someone ticketed for speeding, given that the selected
person is a woman.
Use a 0.05 significance level to test the claim that the percentage of women ticketed for
speeding is less than the percentage of men. Can you conclude that men generally speed more
than women?
11. (Total: 20 points) A study claim that students in two-year college work on average of 20 hours a week.
A teacher wanted to test this claim. She took a sample of 12 students and asked them about the
number of hours they work per week. The following are their responses:
14
25
22
8
16
26
19
23
41
33
21
20
Assume that the number of hours worked by all two-year college students is normally distributed.
a. ( 4 pts) Calculate the (approximate) value of the 85th percentile.
b.
( 4 pts)
Find the percentile rank of 20.
c.
( 4 pts)
Find the sample mean and sample standard deviation of these 12 students.
d.
( 8 pts)
Using a 0.05 significance level, can you conclude that the claim of the study is true?
12. (Total: 10 points) The following are the quiz scores of the seven students in a statistics class.
22
17
29
19
12
25
16
Make a 95% confidence intervals for the population variance. Assume that the population from
which this sample is selected is normally distributed.
13. (10 points) Because of the rapid increase of gasoline prices, many consumers are turned to some
gasoline additives and hoping to save some money. A manufacturer of a gasoline additive claims
that the use of this additive increases gasoline mileage. A random sample of six cars was selected
and these cars were driven for on week without the gasoline additive and then for one week with
the gasoline additive. The following table gives the mile per gallon for these cars without and
with the gasoline additive.
Without
24.6
28.3
18.9
23.7
15.4
29.5
With
26.3
31.7
18.2
25.3
18.3
30.9
Using the 2.5% significance level, can you conclude that the use of the gasoline additive increases
the gasoline mileage? Assume that the population of paired different is approximately normally
distributed.