Areej Jouhar & Hafsa El-Zain Biostatistics BIOS 101 Foundation year

After completing this chapter, you should be able to: ● Describe the Central Limit Theorem and its importance ● Apply sampling distributions for Lecture Goals 2

If a population is Normal with mean μ and standard deviation σ, the sampling distribution of is also Normally Distributed with and 3 If the Population is Normal

● Z-value for the sampling distribution of : Where: = sample mean = population mean = population standard deviation n = sample size 4 Z-value For Sampling Distribution Of

Normal Population Distribution Normal Sampling Distribution (has the same mean) ▪ (i.e. is unbiased ) 5 Sampling Distribution Properties

● For sampling with replacement: As n increases, decreases Larger sample size n Smaller sample size n 6 Sampling Distribution Properties

If the Population is not Normal ▪ We can apply the Central Limit Theorem: ➢ Even if the population is not normal, ➢ …sample means from the population will be approximately normal as long as the sample size is large enough ➢ …and the sampling distribution will have and 7

n↑ Central Limit Theorem As the sample size gets large enough… the sampling distribution becomes almost normal regardless of shape of population 8

Population Distribution Sampling Distribution (becomes normal as n increases) Central Tendency Variation (Sampling with replacement) Larger sample size Smaller sample size If the Population is not Normal Sampling distribution properties: 9

How Large is Large Enough? ● For most distributions, n > 30 will give a sampling distribution that is nearly normal ● For fairly symmetric distributions, n > 15 ● For normal population distributions, the sampling distribution of the mean is always normally distributed 10

Example ● Suppose a population has mean μ = 8 and standard deviation σ = 3. Suppose a random sample of size n = 36 is selected. ● What is the probability that the sample mean is between 8.2 and 9? 11

Example Solution:  Even if the population is not normally distributed, the central limit theorem can be used (n > 30)  … So the sampling distribution of is approximately normal  … with mean = 8  …and standard deviation 12

Example Solution (continued): 13

Summary  Introduced sampling distributions  Described the sampling distribution of the mean  For normal populations.  For not-normal population; Using the Central Limit Theorem. 14

Lecture7 *  Suppose the average length of stay in a chronic disease hospital of a certain type of patient is 60 days with a standard deviation of 15, If it is reasonable to assume an approximately normal distribution of lengths of stay,  find the probability that a randomly selected patient from this group will have a length of stay: 1. Greater than 50 days 2. Less than 30 days 3. Between 30 and 60 days 4. Greater than 90 days 15 Exercise

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ANY QUESTION ?!! The End 19