BUSOBA 2320: Normal Distribution (Include Rec Notes)

PREV CLASS REVIEW

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Statistics/What is this class about?

• Science of learning from data.

• The collection, analysis, interpretation, presentation, and organization of data.

• All about working with data.

Population

• The entire group of interest for a study.

• “Whole group”

• Connected to parameter.

Parameter

• A measurement that describes a population.

• Typically not calculated, they are estimated.

• Only can be calculated with census (population) data.

• Ex. Population proportion (p), population mean (μ/mu), population standard deviation (σ/sigma)

Sample

• A subset of individuals, selected from the population, that we collect data about and analyze.

• Part of the population.

• Connected to statistic.

Statistic

• A measurement that describes a sample.

• Calculated using sample data.

• Can be used to estimate a parameter.

• Ex. Sample proportion (p̂ (read as "p-hat")), sample mean (x̄ (pronounced "x-bar")), sample standard deviation (s), sample size (n)

Mean

• Average of all the data points.

• Found by adding up all the data points and dividing by the sample size (n).

• Can be (μ/mu) or (x̄)

Standard Deviation

• Represents how far the data values are spread away from the mean.

• Is appropriate only for symmetric distributions and when using mean as the measure of center.

• Not resistant to outliers (in other words, is affected by outlier).

• Can be (σ/sigma) or s

Probability

• Long run relative frequency. (Frequency/Total).

• This is for a very large sample size.

• P = _

• P(x) = “P of x” = “_ of x”

• Ex. of rolling an even number, of heads, etc.

Probability Distribution

• The list of all possible outcomes and the probability of each outcome occurring.

Required Properties of Probability Distribution

- All probabilities are between 0 and 1.
- All probabilities are sum to 1.
- Random variable is numerical variable.
- (Additional) Can't have negative probability.

Z Score

• The number of standard deviations away from the mean.

• (observation - mean) / (standard deviation)

EXTRA REVIEW FROM PREV CLASS

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Spread/Variation

Measure of how much the data values vary.

Distribution

Nature or shape over a range of values.

Bell Shape/Curve

• A histogram. Bars of equal width drawn adjacent to each other.

• Symmetric.

• Peak is in the middle.

Frequency

Count of how often an outcome occurs.

Measures of Center

Value that represents the middle or most likely outcome. Ex. Mean

Symmetric Distribution

• Mean = Median = Mode

• Symmetry means shape.

• Traditional bell curve.

• Best measures of center and spread are mean and standard deviation because they are more informative, but not resistant to outliers.

Standard Normal Distribution (Z Distribution)

Normal probability distribution with mean of 0 and standard deviation of 1.

Random Variable

A variable (typically represented by x) that has a single numerical value, determined by chance, for each outcome of a procedure.

VIDEO REVIEW/Some are filled out.

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Statistic VIDEO

• Any quantity that we calculate from data.

• In practice, we usually obtain a _ from a sample and use it to estimate a population parameter.

Parameter VIDEO

• Population model

• not just unknown—usually they
  are unknowable.

• We take a sample and use the sample statistics to
  estimate them.

Mean μ

LOOK AT PREV STATS NOTES

Standard Deviation σ

LOOK AT PREV STATS NOTES

Probability

LOOK AT PREV STATS NOTES

Probability Rule 1

• A probability is a number between 0 and 1.

• If the probability of an event occurring is 0, the event won't occur; likewise if the probability is 1, the event will always occur.

• Even if you think an event is very unlikely, its probability can't be negative, and even if you're sure it will happen, its probability can't be greater than 1.

Probability Rule 2

• The probability of the set of all possible outcomes must be 1.

• Something always occurs, so the probability of some-thing happening is 1. This is called the Probability Assignment Rule:

Probability Rule 3

• The probability of an event occurring is 1 minus the probability
  that it doesn't occur.

• The probability of an event occurring is 1 minus the probability
  that it doesn't occur.

Probability Distribution

LOOK AT PREV STATS NOTES

Z Score

Tells us how many standard deviations a value is from its mean.

NORMAL DISTRIBUTION NOTES

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GIVEN REC NOTES

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Normal Probability Distribution

• Also called Gaussian distributions

• Family of density curves

• Symmetric, bell shape

• Defined by two parameters: the mean, μ (mu) that describes location, and the standard deviation, σ (sigma) that describes spread.

Mean μ (mu)

• Describes location

• the theoretical or POPULATION _

Standard Deviation σ (sigma)

• Describes spread.

• the theoretical or POPULATION _

X ~ N(μ, σ) means and reads as

• X is a Normal random variable

• X is Normally distributed with

  mean = μ and standard deviation = σ

Extra: Means are different, standard deviations are the same.

Bell curves look the same, but are in different places.

Extra: Means are the same, standard deviations are different.

Bell curves are ON TOP of each other, but are narrow or wide.

Extra: Small standard deviation. Ex. 2

• Narrow bell curve.

• Closer to the mean.

Extra: Large standard deviation. Ex. 6

• Wide bell curve.

• Father from the mean.

Z Score

• Measure distance from the mean, as a number of standard deviations.

• = (value - mean) / standard deviation

Extra: Z Score includes value or observation. Standard deviation…

doesn’t.

Empirical Rule, or 68-95-99.7 Rule

• Provides approximate probabilities

Empirical Rule: About 68% of the time observations from a Normal

distribution are within

1 standard deviation (σ) of the mean (μ).

Empirical Rule: About 95% of all observations are within 2  of .

2 standard deviation (σ) of the mean (μ).

Empirical Rule: Almost all (99.7%) observations are within

3 standard deviation (σ) of the mean (μ).

Normal Z-scores →

probability

Normal probability →

Z-scores

Cumulative Area

• P(X ≤ x)

• Graph: A point on the line (x), that goes up and above the bell curve and to the left.

Tail Area

• P(X > x)

• Graph: A point on the line (x), that is within the bell curve and is on its ends.

Interior Area

• Graph: Two points on the line (x). One is the mean (μ) and another is x. The area between these two points that are within the bell curve is the…

Extra: The line (x) can turn into line (z) through the z score equation. On the line (z), the mean will become

0 and x will become z.

Since the Normal random variable is continuous, the point of equality is

irrelevant for probability calculations: P(Z = z) = 0 and P(X = x) = 0.

Total area under the bell curve is

1.

By symmetry, half of the probability is above the mean, half is below the mean. This means that

• P(X ≤ μX) = 0.5000 = P(X ≥ μX)

• Basically, one side is 0.50 and the other side is 0.50.

• P(X ≤ μX) + P(X ≥ μX) = 1

• Basically, if you add both sides together, it will equal 1.

Extra: - z score =

• z score

While several versions exist, one, and only one, standard Normal table is required to answer

ALL probability questions about any Normal random variable.

P(0 < Z < 1.96)

• Graph: On the line (z), everything FROM 0 to the RIGHT until reaching 1.96 is shaded.

• Interior Area

P(-1.96 < Z < 0)

• Graph: On the line (z), everything FROM 0 to the LEFT until reaching -1.96 is shaded.

• Interior Area

P(Z < 1.96)

• Graph: On the line (z), everything FROM 1.96 is shaded to the LEFT, including 0.

• Focus on the UNSHADED tail. Tail Area

P(Z > -1.96)

• Graph: On the line (z), everything FROM -1.96 is shaded to the RIGHT, including 0.

• Focus on the UNSHADED tail. Tail Area. This will be positive.

Extra: z score equation can lead to finding

# of standard deviations away from μ that x is

Extra: Z score equation can also lead to finding

𝑥 = 𝜇 + 𝑧𝜎 𝑜𝑟 𝑥 = 𝜇 − 𝑧𝜎