Normal Probability Distributions Notes

A continuous random variable has infinite possible values represented by an interval on the number line.
The probability distribution for a continuous random variable is known as a continuous probability distribution.
The normal distribution is the most important continuous probability distribution in statistics, characterized by its normal curve.

Mean, Median, and Mode are equal.
The normal curve is bell-shaped and symmetric about the mean.
The total area under the curve equals 1.
The normal curve approaches the x-axis but never touches it as it extends away from the mean.
The graph curves downward between µ−σ and µ+σ and curves upward outside this interval, while the points where the curve changes from upward to downward are called inflection points.
The normal curve can be represented by the equation:
y = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{1}{2} \left( \frac{x - \mu}{\sigma} \right)^2}

A normal distribution can have any mean (µ) and any positive standard deviation (σ).
The standard deviation describes how spread out the data is.
- Example:
- For two curves:
  - Curve A has symmetry at x = 5.
  - Curve B has symmetry at x = 9.
  - Conclusion: Curve B has a greater mean and is more spread out, giving it a larger standard deviation.

The standard normal distribution has a mean of 0 and a standard deviation of 1.
To transform a value to a z-score, use:
z = \frac{Value - Mean}{Standard\ Deviation} = \frac{x - \mu}{\sigma}
After converting to a z-score, utilize the Standard Normal Table to find cumulative area under the curve.

Sketch the standard normal curve and shade relevant areas.
To find area:
- (a) Left of z: use Standard Normal Table.
- (b) Right of z: find corresponding area and subtract from 1.
- (c) Between two z-scores: calculate both areas and subtract smaller from larger.

Using normal distribution, the probability that x falls within a specified interval corresponds to the area under the normal curve for that interval.
- Example:
- Test average = 78, standard deviation = 8. Find probability that a score is < 90:
  z = \frac{90 - 78}{8} = 1.5
  ightarrow P(x < 90) = P(z < 1.5) = 0.9332.

Find the cumulative area that corresponds to a specific z-score.
- Example: Cumulative area of 0.9973 corresponds to a z-score of 2.78.
- For area 0.4170, the z-score is -0.21.

To retrieve a data value x from a z-score:
x = µ + zσ
Example: Monthly electric bill mean = 120, standard deviation = 16, find x for z = 1.60:
- x = 120 + 1.60 * 16 = 145.6.

Sampling Distributions: Formed when samples of size n are drawn repeatedly from a population.
Properties:
1. Mean of sample means µ_{\overline{x}} = µ.
2. Standard deviation of sample means σ_{\overline{x}} = \frac{σ}{\sqrt{n}}.
3. Standard deviation is termed standard error of the mean.
Central Limit Theorem: For n ≥ 30, sample means will follow a normal distribution regardless of population shape.
Example: For 38 bushes with heights mean = 8 feet and standard deviation = 0.7 feet:
- Determine mean and standard error:
- µ_{\overline{x}} = µ = 8
- σ_{\overline{x}} = \frac{0.7}{\sqrt{38}} = 0.11.

The normal distribution approximates the binomial distribution when both np ≥ 5 and nq ≥ 5.
Example: For the probability of students attending college, compute the mean and standard deviation to ensure criteria are met.
Correction for continuity may be necessary when approximating, adjusting binomial to continuous intervals.
Example: For probability ranges, modify them to include continuity by adding/subtracting 0.5.