Continuous Probability Distributions (The Normal Distribution-II) - PowerPoint PPT Presentation

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Continuous Probability Distributions (The Normal Distribution-II). QSCI 381 – Lecture 17 (Larson and Farber, Sect 5.4-5.5). Finding z-scores-I. Yesterday we addressed the question: What is the probability that a normal random variable, X , would lie between x 1 and x 2 .

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Continuous Probability Distributions (The Normal Distribution-II)

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Continuous Probability Distributions (The Normal Distribution-II)

QSCI 381 – Lecture 17

(Larson and Farber, Sect 5.4-5.5)

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 P[X  x1] and P[X  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.

Finding z-scores-II

Area=0.8

X?

What value of x corresponds

to an area of 0.8?

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:

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.

Example-II

-1.64

0

0.674

Find the z-score for each level

(5%, 50% and 75%)

Example-III

• We now apply the formula:

so the maximum lengths are 450.7, 500 and 520.2 mm.

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.

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.

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?

N=10

N=20

N=100

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

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.

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?