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Probability (NORMAL DISTRIBUTION (What is it Many data sets have a…
Probability
NORMAL DISTRIBUTION
What is it Many data sets have a distribution which is bell-shaped when graphed. Data sets with symmetrical bell-shaped graphs are said to have a normal distribution. Each normal distribution is described with two measures:
μ - mean = middle of the bell - shape
σ - standard deviation = how spread out is the bell curve
The total area under the normal curve is 1. Each event corresponds to an interval of values on the horizontal axis. The probability of an event is given by the area under the normal curve over this interval NOTE: The probability is the same whether or not the end points of the interval are included
Z = x - μ / σ
Comment
-Symmetry - skewed (side or not)
- Spread - standard deviation (relation to skew)
- Mean (average) - where is the peak
Show understanding in explanation of these qualities
Standard Normal Tables The table gives the area under the standard normal curve between 0 and z. This area is equal to the probability of a value lying between 0 and z.
Probabilities using Normal Curve Normal probability usually involves variables where the mean is not 0 and/or the standard deviation is not 1. To solve these problems the normal distribution (with the mean and standard deviation) must be converted to a standardised normal distribution. The conversion is done with this formula:
Z = x - μ / σ
Z - number of standard deviations
X - number of interest
μ - mean
σ - standard deviation
Inverse Probabilities using standard tables In reverse to finding a z value between 0 and k, if P(0<z<k) is known, then the table can be used to find the value of k.
- find the given probability in the body of the table
- read off the z-value that corresponds with this value
If the exact probability value is not in the body of the tables:
- find a probability value just below the required probability and find its z-value
- work out the difference that needs to be added onto the lower probability value, and find this value in the difference column (in same row as the z-value already found)
The column number of this difference is added to the end of the z-value.
Standard Normal Distribution The shape of a normal curve depends on the values of mean μ and standard deviation σ. To avoid having to calculate probabilities for each different combination of μ and σ each normal curve is standardised so that the mean becomes 0 and the standard deviation becomes 1.
Solving Inverse Normal problems A probability fact about an unknown x-value is given. Solving such problems involves finding a probability value in the body of the standard normal tables and reading off the associated value.
PROBABILITY TABLES
Relative Frequency (proportion) The number of occurrences of the event is divided by the total number of possible occurrences. This is used when a selection of a group is made at random.
# Conditional Probability Problems involve a restriction on the group from which selections are being made so that the number of possible outcomes is reduced.
Expected Number is the number in a group which would be expected to have a particular feature. Suppose a sample of size n is taken from a population in which the probability of having a certain characteristic is p. Then the expected number in the sample with the characteristic is: EXPECTED NUMBER = np
Risk and Relative Risk When an event is viewed in a negative way the probability of this occurring is referred to as risk. Relative risk is calculated to compare the risk of an event for one group with the event for a second group. This can be written as a ratio, fraction or decimal. Relative risk is the comparison of two probabilities by division and can therefore be taken on any positive value. NOTE: The risk for the second group may be referred to as the base-line risk (the risk for this group forms the denominator of the relative risk factor.
PROBABILITY TREES
Tree Diagrams Are used for displaying sequences of events and their associated probabilities. - each individual set of branches in a tree diagram has a total of 1 - Multiply along the branches to get the probability of event intersections(one event or another) - Add between branches to get the probability of unions of events (one event or another)
# Conditional Probabilities are calculated using a restricted set of outcomes. These may involve tree diagrams
CALCULATORS
Normal Distribution - Finding Probability MORE THAN: Put number in lower column then exp + 10 in upper column LESS THAN: Put number in upper column and -1exp+10 in lower column
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