STAT213
Worksheet #4
1)
Given
p(x) = (1/4)(3/4)(x-1), x = 1, 2, 3, 4,
etc. show that this is a proper probability distribution. Compute the mean and
standard deviation. (see key)
2)
Sam
is an office supplies salesperson. On average, he gets a sale 20% of the time
after meeting with the prospect. If he meets with 4 prospects in a week, what
is the probability he will get exactly 1 sale? (40.96%)
3)
If
he meets with 10 prospects, what is the probability he will get fewer than 2
sales? (37.59%)
4)
If
he meets with 25 prospects in a month, what is the probability he gets at least
2 sales? (97.26%)
5)
In
a year, he usually meets with 300 prospects. If his average commission is
$2500, what is his average commission in a year? What is the maximum number of
sales he can expect 99.7% of the time, rounding to the nearest whole number?
What would be the commission for that? (mean = $150,000; maximum is 81 sales
which has a commission of $202,000)
6)
Suppose
programmers at a software company average 1 bug for every 10,000 lines of code.
What is the probability a program with 10,000 lines of code will be bug free?
(36.79%)
7)
What
is the probability a program with 25,000 lines of code will have at least 2
bugs? (71.27%)
8)
What
is the probability a program with 5,000 lines of code will have at least 1 bug
but no more than 3? (39.17%)
9)
Suppose
a program has 3.5 million lines of code. What would be the mean and standard
deviation of the number of bugs in the program? (mean = 350; standard deviation
= 18.7083)
10)
Suppose
a pizza delivery driver can deliver 4 pizzas per hour on average. What is the
probability a driver can deliver no more than 3 pizzas in an hour? (43.35%)
11)
What
is the probability a driver can deliver at least 5 pizzas in an hour? (37.12%)
12)
Suppose
the pizza restaurant has 2 drivers. What is the probability the 2 drivers can
deliver at least 10 pizzas in an hour? (28.34%)
13)
According
to Chebyshev’s theorem m + 3s represents a reasonable upper limit
on a distribution. If the 2 drivers both work an 8-hour shift, what is the
maximum number of pizzas the restaurant can expect them to deliver? (88)
14)
Based
on historical records, this is the distribution of the number of times people
recycle in a 6-month period:
|
|
0 |
1 |
2 |
3 |
Total |
|
Male |
0.243 |
0.162 |
0.089 |
0.012 |
0.506 |
|
Female |
0.24 |
0.159 |
0.08 |
0.015 |
0.494 |
|
Total |
0.483 |
0.321 |
0.169 |
0.027 |
1 |
What
percentage of how often a person recycles in a 6-month period is explained by the
person’s gender? Based on this, would it be safe to conclude that how often a
person recycles is independent of the person’s gender? (r
2 = 0.0014%)
15)
Based
on historical records, this is the distribution of how many times in a month
people go to the movies as well as engage in other types of entertainment:
|
Movies
| Other |
0 |
1 |
2 |
3 |
Total |
|
0 |
0.004 |
0.01 |
0.02 |
0.018 |
0.052 |
|
1 |
0.048 |
0.25 |
0.104 |
0.016 |
0.418 |
|
2 |
0.208 |
0.184 |
0.048 |
0.004 |
0.444 |
|
3 |
0.052 |
0.022 |
0.006 |
0.006 |
0.086 |
|
Total |
0.312 |
0.466 |
0.178 |
0.044 |
1 |
What
is the mean and standard deviation of the total number of times people go to
the movies and engage in other types of entertainment? (mean
= 2.518; standard deviation = 0.8329)
16) At a video store, the wholesale cost to the store for a DVD is $5. Suppose the number of times a DVD is rented follows this distribution:
|
X |
0 |
1 |
2 |
3 or more |
|
P(x) |
0.02 |
0.12 |
0.29 |
0.19(2/3)x-3 |
a) Verify that this is a proper probability distribution. (see key)
b) Suppose the store wants a minimum average profit of $3.50 per DVD. How much should they charge? Round to the next highest nickel. (charge $2.40)
c) What is the most number of rentals per DVD they can expect at least 8/9 of the time? Round to the nearest whole number. What is the actual probability of no more than that many rentals? (11 rentals: 98.52%)
17) A survey of 2000 households segregated annual household income and annual donations to charity:
|
Charity | Income |
< 25K |
25K to Under 50K |
50K to Under 100K |
100K plus |
Total |
|
< 50 |
412 |
310 |
159 |
0 |
881 |
|
50 to under 200 |
68 |
295 |
204 |
43 |
610 |
|
200 plus |
0 |
125 |
197 |
187 |
509 |
|
Total |
480 |
730 |
560 |
230 |
2000 |
Suppose we divide income into under $50,000/at least $50,000 and charity into under $200/at least $200. With the data arranged this way, what percentage of the variation in charitable donations is explained by income? (18.45%)
From the textbook:
Section 4-2
Q12 (page 184)
Q20 (page 185) Note: Ignore
the hint. It’s easier to use the hypergeometric
distribution.
Section 4-3
Q22 (page 198)
Q28 (page 199)
Section 4-5
Q10 (page 213)