Download Standardization 2(worksheet)

The Standard Normal Distribution
Standardization of Normal Random Variables. If X is normally distributed, its standardization is
1.
What is the distribution of Z?
Suppose that X is normally distributed, with a mean X of 30 and standard deviation of 5.
2. What is the Z-value (that is, the standardized value) of X = 35?
3.
What is the standardized value of X =40?
4.
What is the Z-value of X = 25?
5.
If a value of X is three standard deviations above the mean, what is its Z value? What is the X value?
6.
What is the probability of getting a Z-value that is two standard deviations above the mean?
7.
What is the probability of getting a Z-value that is more than two standard deviations away from the
mean, either above or below?
8.
What is the probability of getting a X value that is two standard deviations above the mean?
9.
What is the probability of getting a X value that is more than two standard deviations away from the
mean, either above or below?
Finding the Z value corresponding to particular probabilities
10. Using Excel, find the value of z0 such that
NORMDIST and trial and error.
11. Find
Give two decimal places. Use
, where z0 is as in #10.
12. Find the value of z0 such that
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A 50 kg sack of flour contains a weight of flour that is normally distributed with mean 51 kg and
standard deviation 0.5 kg.
13. What is the Z-value of a weight of 50 kg?
14. What is the probability of a sack being underweight?
Standardization of Mean from Samples of Size n. By the Central Limit Theorem, the sample means
is normally distributed with mean µ and standard deviation σ. Thus the standardization, S, has the
standard normal distribution, where
This is true no matter what the distribution of X provided the samples are random and n is large
enough (usually above 30). (Quite remarkable!)
15. A sample of 4 sacks of flour has mean 50 kg. What is the Z-value of this mean?
16. What is the probability of a mean of 50 kg or lower?
17. A sample of 25 sacks of flour has mean 50 kg. What is the Z-value of this mean?
18. What is the probability of a mean of 50 kg or lower?
19. A sample of 100 sacks of flour has mean 50 kg. What is the Z-value of this mean?
20. What is the probability of a mean of 50 kg or lower?
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