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 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?


Sampling and Sampling Distributions-IV(Example)

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.


The Central Limit Theorem(Example)


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?


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