1. Central Limit Theorem
Skill 31

2. The 𝒙 Distribution, Given x is normal.
Let x represent the length of a single trout taken at random from the pond. A group of biologists has determined that x has a normal distribution with mean  = 10.2 inches and standard deviation  = 1.4 inches.
What is the probability that a single trout taken at random from the pond is between 8 and 12 inches long?

3. Example 1 – Solution So  = 10.2 and  = 1.4
Therefore, the probability is about that a single trout taken at random is between 8 and 12 inches long.

4. Example– Probability regarding x and x
What is the probability that the mean length of 5 trout taken at random is between 8 and 12 inches long?

5. Example–Solution

6. Example–Solution

7. The 𝒙 Distribution, Given x Follows any Distribution

8. Example–Central Limit Theorem
A certain strain of bacteria occurs in a all raw milk. Let x be milliliter of milk the health department has found that if the milk is not contaminated then x has a distribution that basically normal.
The mean of the x distribution is 2500, and the standard deviation is 300. In a large commercial dairy, the health inspector takes 42 random samples of the milk produced every day.
At the end of the day, the bacteria count in each of the 42 samples is averaged to obtain the sample mean bacteria count .

9. Example–Central Limit Theorem
Assuming the milk is not contaminated, what is the distribution of .
Sample size: n=42 (Greater than 30)

10. Example–Central Limit Theorem
Assuming the milk is not contaminated, what is the probability that the average bacteria count x-bar is between 2350 and 2650 bacteria per milliliter.
Convert to z values

11. Example–Solution

12. Example–Central Limit Theorem.
The mean of the x distribution is 450, and the standard deviation is 30. Suppose we have a random sample of 35. Find the distribution of .
Sample size: n=35 (Greater than 30)
=450

13. Example–Central Limit Theorem.
Find the probability that the average is between 400 and 500.

14. 31 Central Limit Theorem
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15. The 𝒙 Distribution, Given x Follows any Distribution