INTRODUCTION TO STATISTICS & PROBABILITY

Chapter 5: Sampling Distributions

(Part 1)

Dr. Nahid Sultana

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Chapter 5: Sampling Distributions

5.1 The Sampling Distribution of a Sample Mean 5.2 Sampling Distributions for Counts and Proportions

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5.1 The Sampling Distribution of a Sample Mean

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Population Distribution vs. Sampling Distribution

The Mean and Standard Deviation of the Sample

Mean

Sampling Distribution of a Sample Mean

Central Limit Theorem

Sample Mean

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Displays the distribution of customer service call lengths for a bank service center for a month.

There are more than 30,000 calls in this population. (omitted a few outliers)

The population mean is μ = 173.95 sec.

Take a sample of 80 calls from this Population and calculate the mean of these 80 calls.

If we take more samples of size 80, we will get different values of .

This is the sampling distribution of the values of for 500 samples of size 80.

x

x

x

The sampling distribution is roughly symmetric rather than skewed.

The sample means are much less spread out than the individual call lengths.

Sample Mean (Cont…)

Normal quantile plot of the 500 sample means

The distribution is close to Normal.

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Sample Mean (Cont…)

This example illustrates two important facts about sample means:

FACTS ABOUT SAMPLE MEANS 1. Sample means are less variable than individual observations.

2. Sample means are more Normal than individual observations.

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Mean and Standard Deviation of a Sample Mean

Example: Take an SRS of size 36 from a population with mean 240 and standard deviation 18. Find the mean and standard deviation of the sampling distribution of your sample mean.

Now repeat the previous calculations for a sample size of 144. Explain the effect of the increase on the sample mean and standard deviation.

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The Central Limit Theorem

If individual observations have the N(µ,σ) distribution, then the sample mean of an SRS of size n has the N(µ, σ/√n) distribution regardless of the sample size n.

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The Central Limit Theorem (Cont…)

One of the most famous facts of probability theory: When the sample is large enough, the distribution of sample means is very close to Normal, no matter what shape the population distribution has, as long as the population has a finite standard deviation.

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The Central Limit Theorem (Cont…)

Example:

Take an SRS of size 144 from a population with mean 240 and standard deviation 18.

a) According to the central limit theorem, what is the approximate sampling distribution of the sample mean?

b) Use the 95 part of the 68–95–99.7 rule to describe the variability of this sample mean.

c) Suppose we increase the sample size to 1296. Use the 95 part of the 68–95–99.7 rule to describe the variability of this sample mean. Compare your results with those you found before.

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Example: Based on service records from the past year, the time (in hours) that a technician requires to complete preventative maintenance on an air conditioner follows the distribution that is strongly right-skewed, and whose most likely outcomes are close to 0. The mean time is µ = 1 hour and the standard deviation is σ = 1.

Your company will service an SRS of 70 air conditioners. You have budgeted 1.1 hours per unit. Will this be enough?

1== μμx

The sampling distribution of the mean time spent working is approximately N(1, 0.12) :

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=−=>=> zPxP If you budget 1.1 hours per unit, there is a

20% chance the technicians will not complete the work within the budgeted time.

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A Few More Facts

Any linear combination of independent Normal random variables is also Normally distributed. i.e. if X and Y are independent Normal random variables, and a and b are any fixed numbers, then aX + bY is also Normally distributed. In particular, the sum or difference of independent Normal random variables has a Normal distribution. The mean and standard deviation of aX + bY are found as usual from the addition rules for means and variances. Example: Tom and George are playing in the club golf tournament. Their scores vary as they play the course repeatedly. Tom’s score X has the N(110, 10) distribution, and George’s score Y varies from round to round according to the N(100, 8) distribution. If they play independently, what is the probability that Tom will score lower than George .?