Tutorial 5 (Sampling distribution)

X ~ N , 2


 2 
X ~ N  ,
2

n 

 
Standard deviation for X is 
Standard deviation / standard error for X is

n
1)
At a computer manufacturing company, the actual size of computer chips is normally
distributed with a mean of 1 centimeter and a standard deviation of 0.1 centimeter. A
random sample of 12 computer chips is taken. What is the standard error for the sample
mean? (0.029)
2)
A random sample of size 35 is drawn from a normal distribution with mean 30 and
variance 25. What is the probability that
a) The sample mean is at least 28? (0.9911)
b) The sample mean is at most 32? (0.9911)
c) The sample mean is between 29 and 32? (0.8721)
3)
The content of a packet of chocolate drink powder is distributed normally with mean
250 grams and standard deviation of 25 grams. 50 packets were chosen randomly, what
is the probability that the
a) Mean weight is less than 245 grams? (0.0793)
b) Mean weight is between 245 grams and 256 grams? (0.8761)
4)
The average life of a noodle-making machine is 7 years, with standard deviation of 1
year. Assuming that the lives of these machines follow approximately a normal
distribution, find
a) The probability that the mean life of a random sample of 10 such machine falls
between 6.4 years and 7.2 years. (0.707)
b) The probability that the mean life is greater than 8 years. (0.0008)
c) The probability that the mean life is less than 5 years. (0) ~ mean life is never less
than 5 years
d) The value of x to the right of which 15% of the means computed from random
samples of size 9 would fall. (7.345)
5)
The average score of all pro golfers for a particular course has a mean of 70 and a
standard deviation of 3.0. Suppose 36 golfers played the course today. Find the
probability that the average score of the 36 golfers exceeded 71. (0.0228)
6)
The amount of time required for an oil and filter change on an automobile is normally
distributed with a mean of 45 minutes and a standard deviation of 10 minutes. A random
sample of 16 cars is selected. So, 90% of the sample means will be greater than what
value? (41.8 minutes)
7)
The amount of pyridoxine (in grams) per multiple vitamin is normally distributed with
 = 110 grams and  = 25 grams. A sample of 25 vitamins is to be selected.
a) What is the probability that the sample mean will be between 100 and 120 grams?
(0.9544)
b) So, the middle 70% of all sample means will fall between what two values?
(104.818 and 115.182)
8)
A manufacturer of power tools claims that the average amount of time required to
assemble their top-of-the-line table saw is 80 minutes with a standard deviation of 40
minutes. Suppose a random sample of 64 purchasers of this table saw is taken. The
probability that the sample mean will be greater than 88 minutes is __________.
(0.0548)
9)
The amount of bleach a machine pours into bottles has a mean of 36 oz. with a standard
deviation of 0.15 oz. Suppose we take a random sample of 36 bottles filled by this
machine. The probability that the mean of the sample is less than 36.03 is __________.
(0.8849)
10)
The average CPA for faculty of Electrical students is 2.9 with standard deviation 1.0. A
random sample of 100 students is drawn from this population. What is the probability
that the sample mean is smaller than 2.6? (0.0013)

 2
X 1 ~ N  1 , 1

n1

 
2




and

 2
X 2 ~ N  2 , 2

n2

 
2





 1 2  2 2 

X 1  X 2 ~ N  1   2 ,

, standard error for X 1  X 2 is
n1
n2 

 12
n1

 22
n2
11)
A certain type of thread with mean tensile strength of 78.3kg and standard deviation of
5.6kg is manufactured by machine A. Machine B produces a certain type of thread with
mean tensile strength of 77kg and standard deviation of 4kg. A sample size of 40 thread
are taken from machine A and machine B. What is the probability that
a) The difference between mean tensile strength produced by machine A and machine
B is smaller than 2kg? (0.7377)
b) The mean tensile strength produced by machine A is bigger than the mean tensile
strength produced by machine B. (0.883)
12)
An experiment was performed to compare the wear and tear of school shoes produced
by Factory A and B. It is known that the wear and tear of shoes produced by Factory A
is distributed normally with mean 6 months and standard deviation 1 month while the
wear and tear of shoes produced by Factory B is distributed normally with mean 5
months and standard deviation 0.5 month. Random samples of 40 shoes were taken
from both factories. What is the probability that the
a) Difference in the sample mean is larger than 0.5 month? (0.9977)
b) Sample mean of wear and tear of shoes produced by Factory A is larger than the
mean of wear and tear of shoes produced by Factory B. (1)
Continue: Tutorial 5 (Sampling distribution)
Sampling distribution for sample proportion
pˆ 
1)
 pq 
pˆ ~ N  p,

n 

x
n
The National Survey of Engagement shows about 87% of freshmen and seniors rate
their college experience as “good” or “excellent”. Assume this result is true for the
current population of freshmen and seniors. Let p̂ be the proportion of freshmen and
seniors in a random sample of 900 who hold this view. Find the mean and standard
deviation of p̂ .
Solution :
Let p the proportion of all freshmen and seniors who rate their college experience as
“good” or “excellent”. Then,
p = 0.87
and
q = 1 – p = 1 – 0.87 = 0.13
The mean of the sample distribution of p̂ is :
 p̂ = p = 0.87
The standard deviation of p̂ is:
 pˆ 
pq
=
n
0.87 (0.13)
= 0.011
900
2) According to a survey, only 15% of customers who visited the web site of a major retail
store made a purchase. Random samples of size 50 are selected.
a) What is the average of all the sample proportions of customers who will make a
purchase after visiting the web site is _______. 0.15 or 15%.
b) What is the standard deviation of all the sample proportions of customers who will
make a purchase after visiting the web site is ________. 0.05050
c) What proportion of the samples will have between 20% and 30% of customers who
will make a purchase after visiting the web site? 0.1596
d) What proportion of the samples will have less than 15% of customers who will make a
purchase after visiting the web site? 0.5
e) What is the probability that at least 30% of customers who will make a purchase after
visiting the web site? 0.0015
f) 90% of the samples will have less than what percentage of customers who will make a
purchase after visiting the web site? 21.47%
g) 90% of the samples will have more than what percentage of customers who will make
a purchase after visiting the web site? 8.54%
3) A study at a college in the west coast reveals that, historically, 45% of their students are
minority students. Random samples of size 75 are selected. Find
a) the expected percentage of minority students in their next batch of freshmen. 45%
b) the standard error of the proportions of students in the samples who are minority
students. 0.05745
c) the probability that between 30% and 50% of the students in the sample will be
minority students. 0.8033
d) the probability that more than half of the students in the sample will be minority
students.0.1922
e) 95% of the samples will have more than ______% of minority students. 35.55
4) According to an article, 19% of the entire U.S. populations have high-speed access to the
Internet. Random samples of size 200 are selected from the U.S. population.
a) the population mean of all the sample proportions is ______. 19% or 0.19
b) the standard error of all the sample proportions is ______. 0.0277
c) ______ % will have between 14% and 24% who have high-speed access to the
Internet. 92.82
d) ______ % will have between 9% and 29% who have high-speed access to the Internet.
99.97
e) ______ % will have more than 30% who have high-speed access to the Internet.
0.0000
f) ______ % will have less than 20% who have high-speed access to the Internet. 64.06
g) 90 % will have less than _____% who have high-speed access to the Internet. 22.55
h) 90 % will have more than _____% who have high-speed access to the Internet. 15.45
5) Online customer service is a key element to successful online retailing. According to a
marketing survey, 37.5% of online customers take advantage of the online customer
service. Random samples of 200 customers are selected.
a) the population mean of all possible sample proportions is ______. 0.375 or 37.5%
b) the standard error of all possible sample proportions is ______. 0.0342
c) ____ % of the samples are likely to have between 35% and 40% who take advantage
of online customer service. 53.46
d) ____ % of the samples are likely to have less than 37.5% who take advantage of
online customer service. 50
e) 90% of the samples proportions symmetrically around the population proportion will
have between _____% and _____% of the customers who take advantage of online
customer service. 31.87 and 43.13
Sampling distribution of the difference between two proportions


pq 
p q
pˆ 1 ~ N  p1 , 1 1 
pˆ 2 ~ N  p 2 , 2 2
n1 
n2



pq
p q 
pˆ 1  pˆ 2 ~ N  p1  p 2 , 1 1  2 2 
n1
n2 




6) It is known that 30% and 35% of the residents in Taman Sutera and Bandar Mas
subscribe to “New Straits Times” newspaper respectively. If a random sample of 50
newspaper readers from Taman Sutera and 50 readers from Taman Mas were taken
randomly, what is the probability that the proportion of “New Straits Times” subscribes
in Taman Sutera is larger than Bandar Mas?
7) The most popular soccer player is David Beckham. It is known that 60% of the female
soccer fans and 55% of the male soccer fans favor David Beckham over the other soccer
players. A sample of 100 female soccer fans and a sample of 100 male soccer fans were
interviewed at random. What is the probability that
a) proportion of female supporters of David Beckham is larger than the proportion of
male supporters?
b) difference between proportion of female supporters of Beckham and that of male
supporters is at most 0.2?