Download Math 4, Unit 1, Central Limit Theorem

Math 4, Unit 1, Central Limit Thm/Confidence Intervals
Wksht-8.03 – Review: Distributions
Name: _________________________
Date: _____________
I. Let’s say you have a Mathematics 4 teacher that teaches a 90min block. Since her students love math so
much, she plans classes so that each day, 80 to 85 minutes are used for instruction time. Question: what
is the probability that a class will last at most 83 minutes? between 81 and 82.5 minutes?
A visual may help: Graph the
distribution of possibilities.
Notice how the values are spread evenly over the range of possibilities. This distribution is called a
_______________________________. The graph of these results in a _________________ shape.
It is extremely convenient to force the area under the “curve” to 1. Why?
II. Now consider the standard normal curve. What makes it standard?
-3
-2
-1
0
1
2
3
What is the probability that a piece of data will fall less than 1.55? Yeah; z-scores!
Suppose the Morrison Thermometer Company
manufactures thermometers that are supposed to
give readings of 0ºC at the freezing point of
water. It’s not a perfect world, so the
manufacturer guarantees that at freezing, the
thermometers give a mean reading of 0º with a
standard deviation of 1.00º and have a normal
distribution. What is the probability that, at the
freezing point of water, a thermometer will give
a reading of less than 1.58º?
more than -1.23º?
-3
-2
-1
0
1
2
3
-3
-2
-1
0
1
2
3
-3
-2
-1
0
1
2
3
-3
-2
-1
0
1
2
3
-3
-2
-1
0
1
2
3
between -2 & 1.5?
Find the temperature that corresponds with the
95th percentile (the temperature that separates the
bottom 95% with the top 5%).
Sometimes denoted as: P95.
Find the temperatures separating the bottom
2.5% and the top 2.5%.
III. As you know, we can convert from any normal distribution curve (i.e.  ≠ 0,  ≠ 1) to the standard
x
. If a
normal curve using the formula z 

set of data is distributed normally and the mean is
10 with a standard deviation of 3, what is the
probability of a piece of data being less than 12?
The following are SAT (reading and math) data for
graduates in the year 2009 from all high schools in
Georgia:
 = 981
= 215
What percentage earned 1000 or less? Round to
nearest 100th.
1000 or more?
1200 or more?
1500 or less?
What score is at the 95th percentile (P95)? Round to nearest unit.
What score is at the 99th percentile (P99)? Round to nearest unit.
Exercises:
For items 1 – 4, use the problem from I. Find the probability that instruction time ends:
1. in 84.5 minutes or less.
2. in 82 minutes or more.
3. from 82 to 84.5 minutes.
4. less than 81.75 or more than 84.25 minutes
For a standard normal distribution, find the probability of each:
5. less than -0.25
6. less than 0.25
7. greater than 2.33
9. between 0.5 & 1.5
8. greater than -2.33
10. between -2 & -1
11. between -2.67 & 1.28
12. -0.52 & 3.75
13. greater than 3.57
14. greater than 0
15. If the thermometer manufacturer (in II) rejects 6% of all thermometers because the readings are too high
or too low, what degree readings are the cut-offs? Round to the nearest 100th.
For items 16 – 21: IQ scores are normally distributed with a mean of 100 and a standard
deviation of 15. Round to 4 decimal places.
16. Find the probability that someone has an IQ less than 115.
17. Find the probability that someone has an IQ more than 131.5 (Mensa requirement).
18. Find the probability that someone has an IQ between 90 and 110 (normal).
19. Find the probability that someone has an IQ between 110 and 120 (bright normal).
20. At what score is someone at the 90th percentile? nearest tenth
21. At what score is someone at the 45th percentile? nearest tenth
Answers:
1.
3.
5.
7.
9.
11.
13.
15.
17.
19.
21.
0.9
0.5
0.4013
0.0099
0.2417
0.8959
0.0001
0.9821
0.1613
98.1
Answers:
I.
0.2
0.2
85
85
80
or
uniform distribution, rectangular; because probabilities are 0 ≤ P ≤ 1 (there is a correspondence between
area and probability!)
80
II.
mean = 0 and standard dev = 1; 0.9425; 0.8907; 0.9104; 1.645º; less than -1.96 & greater than 1.96;
III.
0.7475; 53.52%; 46.48%; 84.58%; 15.42%; 99.21%; 1335; 1481
Exercises:
1.
0.9
2.
0.6
3.
0.5
4.
0.5
5.
0.4013
6.
0.5987
7.
0.0099
8.
0.9901
9.
0.2417
10. 0.1359
11. 0.8959
12. 0.6984
13. 0.0001
14. 0.5
15. -1.88 & 1.88
16. 0.8413
17. 0.0179
18. 0.4950
19. 0.1613
20. 119.2
21. 98.1