Download IB MATH STUDIES – NORMAL DISTRIBUTION EXAM

AP Statistics – Normal Distribution Practice Exam
Name:______________
DATE:______
POINTS:_____
Draw diagrams for the following. Then calculate the indicated probabilities from the Z
distribution.
1.
between 0.0 and 0.85
Diagram:
Answer: _____________________ (2 marks)
2.
between –0.33 and 0.00
Diagram:
Answer: _____________________ (2 marks)
3.
greater than 0.62
Diagram:
Answer: _____________________ (2 marks)
4.
between –1.57 and –0.89
Diagram:
Answer: _____________________ (2 marks)
5.
less than –0.29
Diagram:
Answer: _____________________ (2 marks)
6.
between –0.33 and 2.40
Answer: _____________________ (2 marks)
Diagram:
Given that Z is normally distributed with =12 and =4, what is the probability of the following?
Please show all work and include a diagram. Give your Answer in decimal and percent form.
7.
Z is between 14.0 and 16.0
Diagram:
Answer: ______________________ (3 marks)
8.
Z is between 10.0 and 12.0
Diagram:
Answer: ______________________ (3 marks)
9.
Z is less than 11.0
Diagram:
Answer: ______________________ (2 marks)
10.
Z is greater than 12.5
Diagram:
Answer: ______________________ (2 marks)
11.
Z is between 11.0 and 14.0
Answer: _______________________ (3 marks)
Diagram:
12.
Find z so that 5% of the area under the standard normal curve lies to the right of z.
Answer: _______________________ (2 marks)
13.
Find z so that 1% of the area under the standard normal curve lies to the left of z.
Answer: _______________________ (2 marks)
14.
Find z so that 95% of the area under the standard normal curve lies between -z and z.
Answer: _______________________ (3 marks)
15.
Find z so that 99% of the area under the standard normal curve lies between -z and z.
Answer: _______________________ (3 marks)
16.
On a practical nursing licensing exam, the mean score is 79 and the standard deviation is 9 points.
a) What is the standardized score of a student with a raw score of 87?
Answer: _______________________ (2 marks)
b) What is the standardized score of a student with a raw score of 79?
Answer: _______________________ (2 marks)
c) Assuming the scores follow a normal distribution, what is the probability that a score selected at
random is above 85?
Answer: _______________________ (2 marks)
17.
In the town of Rockwood, a survey found that the number of hours grade school children watch TV
per week is normally distributed with mean 20 hours and standard deviation 2 hours. If a child is
chosen at random, what is the probability that he or she watches TV
a) Less than 14 hours per week?
Answer: _______________________ (2 marks)
b) More than 22 hours per week?
Answer: _______________________ (2 marks)
c) Between 18 and 20 hours per week?
Answer: _______________________ (3 marks)
18.
Express Courier Service has found that the delivery time for packages is normally distributed with
mean 14 hours and standard deviation 2 hours.
a) For a package selected at random, what is the probability that it will be delivered in 18
hours or less?
Answer: _______________________ (2 marks)
b) What should be the guaranteed delivery time on all packages in order for be 95% sure that the
package will be delivered before this time? (Hint: Note that 5% of the packages will be
delivered at a time beyond the guarantee time period.)
Answer: _______________________ (2 marks)
19.
A Flight for Life helicopter service is available for emergencies occurring 15 to 90 miles from the
hospital. Other emergencies are serviced by ambulance. A study of the service shows that the
response time is normally distributed with a mean of 42 minutes and standard deviation of 8 minutes.
For a random call, what is the probability that the response time will be
a) Between 30 and 45 minutes?
Answer: _______________________ (3 marks)
b) Less than 30 minutes?
Answer: _______________________ (2 marks)
c) More than 60 minutes?
Answer: _______________________ (2 marks)
20.
A factory uses two machines to can fruits. Machine A puts fruit into the cans and Machine B adds
some syrup. The amount of syrup is independent of the amount of fruit in a can.
Machine A is set so that the mean mass of fruit put into each can is 305 grams with a standard
deviation of 2 grams.
(a) Calculate the probability that a can contains less than 300 grams of fruit. [1 mark]
Answer:___________
Machine B is set so that the mean mass of syrup put into each can is 154 grams with a standard
deviation of 1.5 grams.
(b) Calculate the probability that a can contains less than 150 grams of syrup. [1 mark]
Answer:___________
The factory processes 10 000 cans each day.
(c)
(i) How many cans, each day, are expected to contain less than 300 grams of
mark]
fruit? [1
Answer:___________
(ii) How many cans, each day, are expected to contain less than 150 grams of
mark]
syrup? [1
Answer:___________
(iii) Calculate the probability that a can will contain less than 300 grams of fruit and less than
150 grams of syrup. [2 marks]
Answer:___________
(iv) How many cans containing less than 300 grams of fruit and less than 150 grams of syrup
are expected to be produced each day? [2 marks]
Answer:___________
One year there is a shortage of fruit so the manager decides that in order to cut costs he will reduce
the mean mass of fruit to 304 grams and risk a probability that 2.5% of the cans have less than 300
grams of fruit when using Machine A.
(d) What Z-value corresponds to a probability of 2.5% on a standard normal distribution?
[2 marks]
Answer:___________
(e) Calculate the new standard deviation for Machine A. [2 marks]
Answer:___________
Practice Exam Answers:
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
0.3023
0.1293
0.2676
0.1285
0.3859
0.6211
0.1498
0.1915
0.4013
0.4503
0.2902
1.645
-2.33
1.96
2.575
a) 0.89
b) 0
c) 0.2514
a) 0.0013
b) 0.1587
c) 0.3413
a) 0.9772
b) 17.3 hours
a) 0.5812
b) 0.0668
c) 0.0122
a) 0.0062 or 0.62%
b) 0.0038 or 0.38%
c) i. 62 cans
ii. 38 cans
iii. 0.00002356
iv. less than one can
d) -1.96
e) 2.04