Continuous Probability Distributions The Normal DistributionII

1
Continuous Probability Distributions (The Normal
Distribution-II)

• QSCI 381 Lecture 19
• (Larson and Farber, Sect 5.4-5.5)

2
Finding z-scores-I

• Yesterday we addressed the question
• What is the probability that a normal random
  variable, X, would lie between x1 and x2.
• To address this question we found the
  probabilities PX ? x1 and PX ? x2 and
  calculated the difference between them.
• Today we are going to address the inverse of this
  question.
• Find the z-score which corresponds to a
  cumulative area under the standard normal curve
  of p.

3
Finding z-scores-II
Area0.8
X?
What value of x corresponds to an area of 0.8?
4
Finding z-scores-II

• We can use a table of z-scores or the EXCEL
  function NORMINV
• NORMINV(p,?,?)
• Once you have a z-score for a given cumulative
  probability, you can find x for any ? and ? using
  the formula

5
Example-I

• The length distribution of the catch of a given
  species is normally distributed with mean 500 mm
  and standard deviation 30 mm.
• Find the maximum length of the smallest 5, 50
  and 75 of the catch.

6
Example-II
-1.64
0
0.674
Find the z-score for each level (5, 50 and 75)
7
Example-III

• We now apply the formula
• so the maximum lengths are 450.7, 500 and 520.2
  mm.

8
Sampling and Sampling Distributions-I

• So far we have been working on the assumption
  that we know the values for ? and ?. This is
  rarely the case and generally we need to estimate
  these quantities from a sample. The relationship
  between the population mean and the mean of a
  sample taken from the population is therefore of
  interest.

9
Sampling and Sampling Distributions-II
sampling distribution

• A is the
  probability distribution of a sample statistic
  that is formed when samples of size n are
  repeatedly taken from a population. If the sample
  statistic is the sample mean, then the
  distribution is the sampling distribution of
  sample means.

10
Sampling and Sampling Distributions-III(Example)

• Consider a population of fish in a lake. The mean
  and standard deviation of the lengths of these
  fish are 300 mm and 50 mm respectively.
• We now take 100 random samples where each sample
  is of size 10, 20, or 100. What can we learn
  about the population mean?

11
Sampling and Sampling Distributions-IV(Example)
N10
N20
N100
12
Properties of the Sampling Distribution for the
Sample Mean

• The mean of the sample means is equal to the
  population mean
• The standard deviation of the sample means is
  equal to the population standard deviation
  divided by the square root of n.
• is often called the

standard deviation
of the mean
13
The Central Limit Theorem

• If samples of size n (where n ?30) are drawn from
  any population with a mean ? and a standard
  deviation ?, the sampling distribution of sample
  means approximates a normal distribution. The
  greater the sample size, the better the
  approximation.
• If the population is itself normally distributed,
  the sampling distribution of the sample means is
  normally distributed for any sample size.

14
The Central Limit Theorem(Example)
15
Probabilities and the Central Limit Theorem

• The distribution of the heights of trees are not
  normally distributed. We sample 100 (of many)
  trees in a (very large) stand and calculate
  sample mean and sample standard deviation to be
  12.5m and 2.3m respectively.
• What is the standard deviation of the mean?
• What is the probability that the population mean
  is less than 12m?