Download MAT 210 Spring Semester 2004 Answers to the Reviewsheet for the

MAT 210
Spring Semester 2004
Answers to the Reviewsheet for the Final Exam
(1) The following sample gives the area, in square miles, of major islands in the Mediterranean Sea.
1927 3189 3572 1411 9301 9926
Find the mean (4887.7), median (3380.5), variance (14066303.9), and standard deviation (3750.5).
(2) The volume of the gas tanks (in gallons) of 50 selected cars is shown here.
Class
10-13
13-16
16-19
19-22
22-25
Sum
Frequency f
6
24
15
4
1
n = 50
Midpoint Xm
11.5
14.5
17.5
20.5
23.5
2
f Xm
f Xm
69
793.5
348
5046
262.5
4593.75
82
1681
23.5
552.25
2
= 12666.5
f Xm = 785
f Xm
2
f Xm
f Xm
− ( f Xm )2 /n
Find the mean
= 15.7 , modal class (13−16), variance
= 7.0 ,
n
n−1
and standard deviation (2.6).
(3) A disc jockey can select 4 songs from 10 to play in a one-hour show. In how many
ways can this selection be done? (Note: the order of the selection is important.)
(10 · 9 · 8 · 7 = 5040)
(4) A judge is to rank six cookies according to their flavor. In how many ways can this
ranking be done? (6 · 5 · 4 · 3 · 2 · 1 = 720)
(5) There are 12 students who wish to enroll in a particular course. There are only four
seats left in the course.
In how many ways can 4 of the 12 students be selected to
12 · 11 · 20 · 9
= 495
attend the class?
4·3·2·1
(6) A candy store allows customers to select three different candies to be packaged and
mailed.
If there are13 varieties available, how many possibles selections can be made?
13 · 12 · 11
= 286
3·2·1
(7) During a sale at a men’s store, 16 white sweaters, 3 red sweaters, 9 blue sweaters,
and 7 yellow sweaters were purchased. Suppose each customer purchased exactly one
sweater.
(a) Suppose a customer is selected at random, find the probability that he bought:
(i) a blue sweater (9/35), (ii) a yellow or a white sweater (23/35), (iii) a sweater
that is not white (19/35)
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(b) Suppose two different customers are randomly selected, find the probability that:
(i) the first customer bought a blue sweater, the second customer bought a red
sweater (9/35) · (3/34) = 27/1190, (ii) both customers bought a yellow sweater
(7/35) · (6/34) = 42/1190.
(8) At Tyler’s Tie Shop, Tyler found the probabilities that a customer will buy 0,1,2,3, or
4 ties, to be as shown.
0
number of ties
probability
0.30
1
2
3
4
0.50 0.10 0.08 0.02
Find the mean (0.30 · 0 + 0.50 · 1 + 0.10 · 2 + 0.08 · 3 + 0.02 · 4 = 1.02), variance
(0.30 · 02 + 0.50 · 12 + 0.10 · 22 + 0.08 · 32 + 0.02 · 42 − (1.02)2 = 0.8996), and standard
deviation (0.9485) of the random variable that describes the number of ties that a
customer will buy.
(9) The average number of miles a mail carrier walks on a typical route is 5.6. The standard
deviation is 0.8 miles. Assuming the random variable describing the number of miles a
carrier walks is normally distributed, find the probability that a randomly chosen mail
carrier walks: (i) between 5 and 6 miles (0.4649), (ii) less than 4 miles (0.0228), (iii)
more than 6.3 miles (0.1894).
(10) The average weight of a checked item of an airline’s passenger is 45 pounds, with a
standard deviation of 2 pounds. Suppose the weight is normally distributed. The
airline wants to set a weight limit so that 15% of the checked items will be over the
limit (and so subject to additional charges). Find the weight limit. 47.08 pounds
(11) The average weight of 60 randomly selected compact automobiles was 2627 pounds.
The sample standard deviation was 400 pounds. Find the 99% confidence interval of
the true mean weight of compact automobiles. 2494 < µ < 2760
(12) A recent study of 25 commuters showed that they spent an average of $18.23 on public
transportation per week. The standard deviation of the sample was $3.06. Find the
95% confidence interval of the true mean. Assume the variable is normally distributed.
16.97 < µ < 19.49
(13) The standard deviation of the diameter of 28 oranges was 0.34 inch. Find the 90%
confidence interval for the true standard deviation for the diameters of the oranges.
Assume the variable is normally distributed. 0.28 < σ < 0.44
(14) A high school counselor wishes to test the theory that the average age of the dropouts
in her school district is 16.3 years. She samples 32 recent dropouts and finds that their
mean age is 16.9 years. At the significance level α = 5%, should she reject or not reject
the theory? The standard deviation of the population is 0.3 years.
H0 : µ = 16.3 (theory), critical region: z < −1.96 or z > 1.96, test statistic: z = 11.3,
decision: reject H0
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(15) A recent study reported that the average age of murder victims in a small city was
less or equal to 23.2 years. A sample of 18 victims had a mean of 24.6 years and a
standard deviation of 2.7 years. At the significance level α = 10%, does the sample
yield enough information to reject the results of the study? Assume the variable is
normally distributed.
H0 : µ ≤ 23.2 (study), critical region: t > 1.333, test statistic: z = 2.1998, decision:
reject H0
(16) A study is conducted to determine the realtionship between a driver’s age and the
number of speeding tickets he or she has over a one-year period. The data is shown
here.
driver’s age, x
speeding tickets, y
16 17 18 23 24 27
3 2 5 0 1 1
(a) Find the sample correlation coefficient. r = 0.679
(b) Test the significance of the correlation coefficient at α = 0.05. Use the t-test
for the null hypothesis H0 : ρ = 0. H0 : ρ = 0, critical region: t < −2.776 or
z > 2.776, test statistic: t = 1.8497, decision: do not reject H0
(c) Determine the equation of the regression line y = a + bx. y = 7.6914 − 0.2732x
(d) Explain how the decision in part (b) influences whether the regression line can
be used for modelling the relationship between a driver’s age and the number of
speeding tickets.
Since we do not reject the hypothesis that x and y are not linearly related, the
regression line should not be used for modelling and prediction.
(17) A quality control engineer wishes to determine whether the number of defective items
manufactured during the week is the same for each day. A week is selected at random,
and the number of defective items produced each day is shown here.
Day
Monday
Tuesday
Wednesday
Thursday
Friday
Defective Items
40
16
23
19
32
At the significance level α = 5%, use the χ2 -test for goodness of fit to test whether the
number of defective item is the same for each day.
H0 : E1 = E2 = . . . = E5 = 26, critical region: χ2 > 9.488, test statistic: χ2 = 15,
decision: reject H0 , there are significant differences in the number of defective items
among the days of the week
(18) A survey of 164 state representatives is conducted to determine whether their opinion
(approve/disapprove/no opinion) on a bill are related to their party affiliation. The
following table of data is obtained.
3
Party
Republican
Democrat
Independent
Approve Disapprove
27
15
43
18
9
15
No Opinion
13
12
12
At the significance level α = 1%, use the χ2 -test for independence to test whether the
opinion is independent of party affiliation.
H0 : opinion is independent of party affiliation, critical region: χ2 > 13.277, test
statistic: χ2 = 10.904, decision: do not reject H0 , at α = 1%, we can assume that the
opinion is independent of party affiliation
(19) In order to find out whether there is a significant differnce in the average number of
spelling errors made by the students of 3 composition instructors, an ANOVA test
is conducted. It gave a p value of 0.0713. Would you reject or not reject the null
hypothesis at the (i) 10% significance level (reject, since p-value < significance level),
(ii) 5% significance level (do not reject, since p-value ≥ significance level)?
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