1. PROBABILITY AND STATISTICS
WEEK 9
Onur Doğan

2. The sampling distribution of the sample statistics
Onur Doğan

3. The sampling distribution of the sample statistics
Consider a population of N elements from which we can obtain the following distinct data: {0, 2, 4, 6, 8}.
Form samples of size 2 for this population.
Define their means and figure the bar chart of the means.
Define the sampling distribution of the sample ranges and figure bar chart.
Onur Doğan

4. The Central Limit Theorem
The mean is the most commonly used sample statistic and thus it is very important. The central limit theorem is about the sampling distribution of sample means of random samples of size n. Let us establish what we are interested in when studying this distribution: 1) Where is the center? 2) How wide is the dispersion? 3) What are the characteristics of the distribution? The central limit theorem gives us an answer to all these questions.
Onur Doğan

5. The Central Limit Theorem
Let µ be the mean and σ the standard deviation of a population variable. If we consider all possible random sample of size n taken from this population, the sampling distribution of sample means will have the following properties:
c) if the population is normally distributed the sampling distribution of the sample means is normal; if the population is not normally distributed, the sampling distribution of the sample means is approximately normal for samples of size 30 or more. The approximation to the normal distribution improves with samples of larger size.
Onur Doğan

6. The Central Limit Theorem
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7. The Central Limit Theorem
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8. The Central Limit Theorem
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9. Example
Consider a normal population with µ=100 and σ=25. If we choose a random sample of size n = 36, what is the probability that the mean value of this sample is between 90 and 110? In other words, what is P(90 < x < 110)?
Onur Doğan

10. Example
The average male drinks 2L of water when active outdoor s(with standard deviation of 0,7 L). You are planning a full day nature trip for 50 men and bring 110 L of water. What is the probability that you will run out?
Onur Doğan