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Chapter 6: Probability - a fraction or a proportion of all the possible …
Chapter 6: Probability - a fraction or a proportion of all the possible
outcomes, identified as A, B, C, D, and so on.
RANDOM SAMPLING
Simple Random Sampling - each individual in the population has an equal chance of being selected.
Independent Random Sampling - if more than one individual is being selected, the probability must stay constant from one selection to the next.
Must sample with replacements.
FREQUENCY DISTRIBUTIONS
EXAMPLE - population
N
= 10 with values 1,1,2,3,3,4,4,4,4,5,6. random sample
n
=1; probability of score greater than 4?
p
(
X
> 4) = 2/10
TOOL - UNIT NORMAL TABLE (UNT) with 4 columns for z-Scores, body, tail, and between
M
and
z
EXAMPLES - find probabilities and proportions associated with z=Scores
What proportion of the normal distribution corresponds to z-score values greater than
z
= 1.00? Which part? Tail (Column C) for 1.00.
As a proportion, for a normal distribution, what is
the probability of selecting a z-score value greater than
z
= 11.00?
NOTE - Pay attention to < or >
Find
probabilities for specific X values
NORMAL DISTRIBUTION
FORMULA - Step 1: Calculate the z-Score by
z = X
- μ / σ (z-score is X minus mean, divided by sd.) Step 2: Write with z-Score, P(Z > z-score) = ?
Although the z-score values change signs (1 and 2) from one side to the other,
the proportions are always positive.
Normal Distribution is symmetrical, with the highest frequency in the
middle and frequencies tapering off as you move toward either extreme
FORMULA - Probability of A =
p
= # of outcomes A / total number of possible outcomes ie. coin...
p
(heads)=1/2
Can also use to analyze for percentiles and percentile ranks (chpt 2)
NOTE - Pay attention to - (negative) z-scores. Ex. what
proportion of the normal distribution is contained in the tail beyond z = -0.50? or
p(z
< -0.50) = ?
KEY - left is identical to right; solve for right side
z
= +0.50. Right tail beyond .50 (Column C)
Step 1: Transform the
X
values into
z
-scores. Step 2: Use the UNT look up the proportions corresponding to the z-score values.
EXAMPLE -
It is known that IQ scores form a normal distribution with μ = 100 and σ = 15. Given this information, what is the probability of randomly selecting an individual
with an IQ score of less than 120?
Step 1: Change X to x-score: z = X - μ / σ, so 120-100 / 15 = 20/15 = 1.33
Step 2: Look up z-score in UNT. p(X<120) = p(z<1.33) = 0.9082 or 90.82%
Find the Probability of Scoring Between
NOTE - can be solved using the proportions of columns B and C (body and tail), they are often easier to solve with the
proportions listed in column D.
Average speed of μ = 58 miles per hour with a standard
deviation of σ = 10. what proportion of the cars are traveling between 55 and 65 miles per hour?
Step 1: Do z = X - μ / σ for each number.
X
=55 is -0.30;
X
=65 is 0.70.
Step 2: 2 sections, area left of mean between mean and z= -0.30 and right of mean between mean and z=+0.70. Using D, 0.1179 and 0.2580, add these two together for the total proportion.
Answer:
p(z
< 1.00) = 0.1587 (or 15.87%)