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- Normal Distribution and Z-score

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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 $n \geq 30$

$\cdot$ $\mu_{\overline{x}}=\mu$

$\cdot$ $\sigma_{\overline{x}}=\frac{\sigma}{\sqrt{n}}$

Central Limit Theorem:

$Z=\frac{\overline{x}-\mu_{\overline{x}}}{\sigma_{\overline{x}}}=\frac{\overline{x}-\mu}{\frac{\sigma}{\sqrt{n}}}$

Typically $n \geq 30$

- Introductiona)The distribution of sampling means is normally distributedb)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%?

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