# Central limit theorem

### Central limit theorem

#### Lessons

The distribution of sampling means is normally distributed
$\cdot$ $\mu_{\overline{x}}=\mu$
$\cdot$ $\sigma_{\overline{x}}=\frac{\sigma}{\sqrt{n}}$

Central Limit Theorem:
$Z=\frac{\overline{x}-\mu_{\overline{x}}}{\sigma_{\overline{x}}}=\frac{\overline{x}-\mu}{\frac{\sigma}{\sqrt{n}}}$
Typically $n \geq 30$
• Introduction
a)
The distribution of sampling means is normally distributed

b)
Formula for the Central Limit Theorem

• 1.
Comparing the Individual Z-Score to the Central Limit Theorem
A population of cars has an average weight of 1350kg with a standard deviation of 200 kg. Assume that these weights are normally distributed.
a)
Find the probability that a randomly selected car will weigh more than 1400kg.

b)
What is the probability that a group of 30 cars will have an average weight of more than 1400kg?

c)
Compare the two answers found in the previous parts of this question.

• 2.
Applying the Central Limit Theorem
Skis have an average weight of 11 lbs, with a standard deviation of 4 lbs. If a sample of 75 skis is tested, what is the probability that their average weight will be less than 10 lbs?

• 3.
Increasing Sample Size
At the University of British Columbia the average grade for the course "Mathematical Proofs" is 68%. This grade has a standard deviation of 15%.
a)
If 20 students are randomly sampled what is the probability that the average of their mark is above 72%?

b)
If 50 students are randomly sampled what is the probability that the average of their mark is above 72%?

c)
If 100 students are randomly sampled what is the probability that the average of their mark is above 72%?