Stat 202: Lecture 15 (covers pp. 166-169)

Nathan VanHoudnos
10/29/2014

Agenda

  1. Checkpoint #16 comments
  2. Lecture 15 (covers pp. 166-169)

Checkpoint #16

To fill in.

Agenda

  1. Checkpoint #16 comments
  2. Lecture 15 (covers pp. 166-169)

Stat 202 focuses on

Population Parameters

Continuous

  • Population mean: \( \mu \)
  • Population standard deviation: \( \sigma \)

Categorical

  • Population proportion: \( p \)

Sample Statistics

Continuous

  • Sample mean: \( \bar{x} \)
  • Sample standard deviation: \( s \)

Categorical

  • Sample proportion: \( \hat{p} \)

Height of Adult males

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Assume that the height of adult men is normally distributed with a mean of 69 in. and a standard deviation of 2.8 in.

Height of Adult males

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Assume that the height of adult men is normally distributed with a mean of 69 in. and a standard deviation of 2.8 in.

  • Randomly select 15 men from the population and measure their heights:
67.2 69.5 66.7 73.5 69.9 66.7 70.4 71.1 70.6 68.1 73.2 70.1 67.3 62.8 72.1

Height of Adult males

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Assume that the height of adult men is normally distributed with a mean of 69 in and a standard deviation of 2.8 inches.

  • Randomly select 15 men from the population and measure their heights:
  • Make a histogram of those randomly selected 15 men.

Sample mean of Adult Males

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The population mean

  • always 69 inches.

Each sample mean

69.3 69.7 68.5 70.2 68.5 69.2

Sample 15 men 1,000 times

69.3 69.7 68.5 70.2 68.5 69.2 70.6 67.9 69.1 68.7 67.8 67.6 69.3 70.6 69.5 69.9 70.2 69.1 69.2 68.1 69.1 69.7 69.9 68.5 67.7 69.8 69.6 68.2 69.5 68.2 68.9 69.5 69.3 69.3 69.6 69.3 69.8 68.5 68.6 68.2 69.8 70.4 68.5 67.6 69.1 69.7 68.4 70.2 68.7 68.6 68.7 68.8 69.2 70 68 68.9 69.7 68.6 69.6 68.7 68.4 68.8 68.5 68.4 68.1 68.5 68.7 69.6 68.2 68.7 68.4 69.9 70.3 68.6 69.7 68.3 69.2 68.5 68.1 68.8 69 68.3 69.5 68.5 67 69.1 68 69.1 68.4 69.9 69.5 68.4 70.5 69 69 69.4 69.1 68.9 67.9 69.1 69.4 70.3 68.6 69.8 68.8 69.1 68.8 69.1 69.2 70.9 68.2 68.9 68.8 68.8 69.2 70 68.6 68.4 68.3 69.6 69.2 68 69.4 69.5 69.1 69 69.3 70.1 68.8 69.1 69.2 69.1 69.4 69.8 69.4 69 70.4 68.5 68.3 70.8 69.1 70 70.3 67.9 70.4 67.9 68.6 69.6 67 68 68.6 68.4 68.4 69 69 68.8 70.5 68 67.8 69.6 70.7 68.7 69 69.9 69.1 69.6 68.2 69.6 69 70.2 69.4 69.6 70.6 70.7 70.2 69.7 69.7 69.3 69.3 69 68.9 70.4 68.8 69.5 69.3 68.5 68.9 68.3 69.5 69.6 70.5 69 69.5 68.9 68.5 69.3 70.1 69 68.4 68.8 68.5 68.2 68.6 68.2 69.5 69.3 67.4 70.4 69.2 69.4 69 69.5 68.1 68.7 69.3 70.4 68.1 68.1 68.2 67.5 68.6 69.3 69.3 68.8 68.5 68.4 68.8 69.1 69.1 69.2 68.4 69.8 69.6 68.7 68.5 68.8 68.6 68.3 69.2 69.5 68 69.1 68.7 67.7 68.6 69.9 69.5 68 70.6 68.9 68 69.7 68.8 69.7 69.6 68.4 69.1 69.6 69 68.8 68.2 68.9 68.7 69.9 67.9 70 69.1 68.9 68.4 68.7 68.6 69.1 68.3 68 68.8 69 69.4 68.4 70 68.5 69.9 68.8 70.4 69.4 69.1 70.2 69 68.8 69.2 69.5 68.5 68.9 70 69.3 68.9 69.2 67.7 68.9 68.2 70.7 68.8 69 69.4 69.6 69.4 68.7 68.5 67.8 68.8 68.9 69.2 69.1 69 67.6 68.5 68.5 69.4 68.7 68.8 69.1 69.9 68.7 70.3 69.2 68.9 69.6 68.8 69.7 69 69.4 68.9 69.7 68.4 69.1 66.8 69.5 68.4 70.3 68.9 68.5 69.4 69.7 67.5 69.8 69.5 68.4 68.8 69.2 69.1 69.7 70.4 68.4 67.9 69.6 67.3 69.3 69.1 68.8 68.2 68.9 69.4 69.6 68.7 68.7 69.6 69.1 68 68.6 69 68.7 69.2 69.6 67.4 68.5 68.8 69.3 69.7 70 68.6 68.5 68.3 69.2 67.6 68.3 69.2 68.4 69.9 68.9 68.2 69.3 68.2 69.4 68.7 70.5 69.1 69.3 68 68.3 68.4 68.2 68.8 70.3 68.9 68.6 69.4 70.3 68.6 68.3 69.9 69.3 67.9 69.6 69.7 67.9 68.8 69.1 68 67.4 69.7 68.4 69.8 69.2 69.1 68.1 68.5 69.7 69.3 69.7 69.3 69.5 67.4 69.2 70.5 69.6 69.3 69.9 68.5 70.3 69.5 69.8 68 68.4 68.6 68.4 66.9 68.9 68.2 68.6 68.9 69.1 71.1 68.5 69.2 69.6 70.5 69.3 68.9 68.7 69.7 70 69.2 69.1 69.7 68.9 69.4 68 69.2 68.1 71 68.9 69.1 67.9 69 67.5 69.1 68.8 68.2 68.7 69.4 68.3 69.2 68.4 68.2 68.7 67.1 69.3 69.1 69 69.1 68.4 69.4 69 68.4 68.7 68.8 68.7 71 68 67.9 69.1 69.4 69 69.7 70.7 67.2 69.7 69.3 67.6 69.1 69 67.7 69.2 69 70.4 69.4 69.8 68.1 68.7 68.4 68.2 68.5 69.9 68.4 68.8 68.5 69.3 68.9 69.5 68.5 69.5 67.8 70 68.5 68.4 69.6 66.9 68.7 69.5 69.7 68.9 68.5 69.3 69.2 68.6 68.6 67.8 68.3 70.2 69.4 70.6 69.4 67.9 69.1 68.9 69 68.6 69 69 70.2 68.7 69.3 68.2 68.5 68 69.6 69.6 67.4 69.8 69.1 70.3 67.7 69.5 70.2 69.8 68.5 70.6 68.3 69.5 68.9 68.3 69.1 69.7 69.7 68 69.5 68.4 69.7 69.6 68.4 68.7 69.2 68.8 69.8 67.9 70.2 68.9 68.1 69.7 68.8 67.7 68.5 67.8 70.4 70 68.8 68.6 69 69 67.7 68 67.9 68.9 69.4 68.2 68.6 69.6 70.2 68.2 69.1 68 69 69.3 69.6 68.6 69.4 68.5 68.5 69.6 69.2 69.6 68.8 68.3 69.1 69.8 69.6 69 69.2 70 69.1 68.8 69.6 69.4 69.5 69.7 67.6 68.5 69.2 68.8 69.2 69.9 70.1 70 69.8 69 69.1 70 68.6 69.6 70.4 69.2 69.4 69.3 68.9 69.5 69.5 68.8 69.4 69 68.1 68.9 68.6 69.2 68.3 69 67.8 69.9 69.8 68.4 69.1 69.9 67.6 69.2 68.8 69.4 69.4 68.2 70.4 68.6 67.7 69 68.4 68.2 69.1 69.7 69.2 68.1 69 69.7 68.8 68.3 68.7 68.4 67.9 69.3 69.7 69.3 69.2 69.5 69.7 69.3 68.8 69 67.9 70.1 69.8 69.6 69.5 69.5 69.2 67.8 68.7 70.1 67.5 69.8 69.1 67.9 70.1 67.5 68.7 68.8 69.9 70.3 69.4 68.3 69.9 69.5 68.7 68.7 68.3 69.6 69.2 69.6 69.6 69.9 68.7 69.7 68.6 68.6 69.5 69.4 67.6 68 68.6 68.7 68 69.6 68.3 69.5 68.2 69.1 68.7 68.5 69.5 69 68.8 70 69.6 69 67.8 70 68.6 67.8 69.1 69.1 68.3 69.6 69.6 70.3 68.8 67.9 69.5 68.6 68.7 69.1 69.1 68.4 69.7 69.6 68.1 70 68.8 68.1 69 68.3 68.7 69.5 68.8 69.9 69.5 68.6 68.5 69.8 69 69.4 68.3 68.8 68.2 69.1 68.6 68.7 68.8 68.4 69.4 69.2 68.5 68.9 70.2 68.6 69.3 67.3 69.4 68.5 68.6 70.3 68.9 67 68.4 69.4 68.4 70.4 69.3 69.3 69.9 69.4 69.5 69.6 68.5 69.1 68.7 68.1 68.6 69.7 67.9 70.1 68.5 70.7 69.7 69.2 69.4 69.6 69.7 68.5 68.1 70.1 69.7 68.1 69.1 69.4 69.5 68.4 69.2 69.2 69.5 70.4 69.4 69.4 69.2 68.3 68.4 68.1 69.4 69.2 67.5 67.3 68.7 69.9 69.1 69.3 70.2 68.9 68.4 67.9 68.4 70.1 68.7 69.5 69.5 68.6 69.3 70.3 68.8 69 67.7 67.3 69 68.5 69.3 69.1 68.5 66.5 69.2 69.2 69.4 69.3 69.6 67.4 68.7 69.3 69.3 68.5 69.8 69.4 69.5 69.2 68.5 68.6 68.6 69 68.8 69.8 69.5 69.1 68.8 68.6 67.4 68.7 66.6 69.3 68.3 68.4 68 69.4 68.1 68.2 68.2 68.7 68.4 69.3 70 69.1 69.1 68.6 68 68.7 68.8 69 68 69.4 68.4 68.4 68.8 68 67.6 69.5 68.2 69 66.9 70.4 68.8 68.6 68.5 68.4 70.5 67.6 68.8 69 68.5 68.7 67.6 68.6 69 68.5 67.5 68.1 68.5 69.9 69.8 69.3 69 69.2 69.3 69.7 69.9 69 69.9 69.3 68.6 69.2 68.7 70.2 69.3 69.1 68.3 68.7 69.5 69 69.2 68.9 68.4 67.8

A histogram of the sample means

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  • Randomly select 15 men from the population and measure their heights.
  • Do this 1,000 times
  • The distribution of the sample mean is close to the population mean (69 inches)

What if we took 50 men instead?

Average of 15 men

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Average of 50 men

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Approximately normal (which?)

If \( X \) is normally distributed with mean \( \mu \) and variance \( \sigma^2 \)

\[ X \sim N( \mu, \sigma^2) \]

then the mean of a sample of \( n \) indepedent draws from \( X \)

\[ \bar{x} = \frac{1}{n} \left[x_1 + x_2 + \dots x_n \right] \]

is also normally distributed

\[ \bar{x} \sim N\left( \mu, \frac{\sigma^2}{n} \right) \]

If X is normal, then

\[ \begin{aligned} \bar{x} & = \frac{1}{n} \left[x_1 + x_2 + \dots x_n \right] & & & \bar{x} & \sim N\left( \mu, \frac{\sigma^2}{n} \right) \end{aligned} \]

Proof

  1. The sum of two (or more) independent normal random variables is also normally distributed.
  2. A normal distribution is defined by its mean and variance.

\[ \bar{x} \sim N\left( E[\bar{x}], \text{Var}[\bar{x}] \right) \]

so find \( E[\bar{x}] \) and \( \text{Var}[\bar{x}] \) (on the board.)

Example 2: Radioactive Decay

a

(for scale)

Bananas are radioactive.

  • In one gram of banana, 31 atoms of potassium-40 will decay every second.
  • In a 150 gram banana \[ 150 * 31 = 4,650 \] decay events happen per second.

Example 2: Radioactive Decay

Time between decays