Download Math Methods - cloudfront.net

Math Methods
Chapter 13, 16-17
Name_______________________
Statistics & Probability 2 Supplemental
Problems 
1. Draw a frequency diagram using intervals of five beginning at 0 (show
the frequency table):
12 14 24 22 18 7 4 8 19 21 13 3 18 1 10 8 6 11 14
20 5 2 15 19 21 7 8
2. Find the mean (no 1-var stat)
3. Using your calculator find the
mean, median and mode correct to
three significant figures:
12.8 9 14 2.6 7.2 6.8 6.8
5.9 12.8 4.8 19.4
Score
50-54
55-59
60-64
65-69
70-74
75-79
Freq
3
8
7
2
4
3
4. Find the quartiles and draw a box-and-whisker plot:
83 62 77 14 82 53 22
19
43
91
59
5. Find the standard deviation (do not use 1-var stat) correct to three
significant figures:
4 6 2 5 8 1 9 4
6. Find the standard deviation (do not use 1-var stat) of #2.
7. Consider the following set of data:
12, 4, 9, 10, 12, 13, 15, 11, 12, 15, 14, 8, 9, 10, 12, 9, 10, 16,
14, 13, 12, 15, 9, 10, 12
a. Construct a cumulative frequency table and curve
b. From the curve, determine the median and quartiles
c. Calculate the interquartile range
8. Using a normal table, find
i. p z  3.32
ii. p 0.65  z  2.76
b
g
b
g
b
g
.
iii. p z  118
9. A brand of tinned baked beans have a mean contents of 345 grams per
tin with a standard deviation of 2.8 grams. Assuming that the
distribution is normal, what percentage (to the nearest whole number)
of the tins contain:
i.
less than 347 grams
ii. more than 345.5 grams
iii. between 343 and 346 grams
If the lightest 1% of the cans are considered to be underweight:
iv. find the weight below which the cans are considered underweight
478152692
10.
As a result of a certain random experiment, the events A and/or
B may occur. These events are independent, and P A  0.5; P B  0.2
bg
bg
a. Let X denote the random variable which counts how many of the
two events occur at a given time. Thus, for example, X  0 if
neither A nor B occur. Find P X  x for x  0, 1, 2
b. Find the mean and variance of X
b g
11.
A certain brand of soft-drink is sold in so-called ‘litre’
bottles. In fact, the amount of liquid in each bottle (in litres) is
a normally distributed random variable with mean 1.005 and standard
deviation of 0.01.
a. Find the proportion of soft-drink bottles containing less than 1
litre
b. If I buy four bottles, find the probability that all four
contain less than 1 litre
12.
A bag contains 3 red and 2 black marbles. Let X be the number
of marbles withdrawn (at random), one at a time without replacement,
until the first black marble is drawn.
a. Explain why X cannot take any value greater than 4
b. Specify the probability distribution of X
c. Find
i. E X
ii. E X 2
iii. E 2 X  1
bg
ch
b g
13.
A discrete random variable X may take the values 0, 1, or 2. The
k
probability distribution of X is defined by P X  x  . Find K
x!
b g
14.
The cross-sectional area of a rod produced by a machine is
normally distributed with a standard deviation of  cm 2 and a mean of
4.0 cm2 .
a. If the proportion of rods with cross-section area of less than 3.0
cm2 is 0.04, evaluate  .
b. If all rods with cross-sectional area of less than 3.0 cm2 are
rejected, what is the probability of an accepted rod having a
cross-sectional area greater than 5 cm2 ?
15.
The mean diameter of bolts from a machine can be adjusted so
that the proportion of bolts greater than 1.00 cm is 0.05, and the
proportion less than 0.90 cm is 0.01. Assuming the distribution of
the bolt diameters to be Normal, find the mean and the standard
deviation of the diameter.
478152692
16.
Of 5 cards, 3 are labeled with a 1, the others with a 2. Three
cards are drawn at random from the five cards, observed, then
returned to the pack. This process is repeated a second time. If X
denotes the number of times two 1’s and a 2 are drawn,
a. Find the probability of two 1’s and a 2 on the first draw.
b. Find P X  x , for x  0, 1, 2 for the two draws
c. Calculate the mean and variance of X
d. Calculate P X  1 X  0
b g
c
h
17.
A tennis player find that he wins 5 out of 7 games he plays.
he plays 7 games straight, find the probability that he will win
If
a. Exactly 3 games
b. At most 3 games
c. All 7 games
d. No more than 5 games
e. After playing 30 games, how many of these would he expect to
win?
18.
If X~N(50, 25) find
a. P 25  X  45 X  40
c
b g
h
b. c if P X  c .9553
19.
the end
ANSWERS:
ANSWERS:
1.
2.
5.
8a. .0005
8b. .7393
.4020
9iv. 338
10b.
12b.
Mean: .7
478152692
62.9
2.57
8c. .119
10a.
Variance: .41
11a.
3.
6.
9i.
x  9.28; med  7.2; mode  6.8
7.70
.7625
.3085
11b.
9ii. .4291
4.
x9iii.
P(x)
0
.4
1
.5
P(x) only
.0091 x 12a.
2 3
.1 reds
2
1
5
2
3
10
3
1
5
4
1
10
12ci. 2
15.
16a.
17b.
12cii. 5
Mean=
.959
12ciii. 3
x
P(x)
16b.
4
0
.108
25
41.5
478152692
12
25
2
9
25
k
2
5
14a.
.5712
14b.
.04
SD= .025
16c.
3
5
1
13.
17c.
.095
mean 1.2, var .48
17d.
.64
17e.
16d.
21
9
21
18a.
17a.
.085
.1391
18b.