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Quantitative Methods - Probabilities and the Normal Curve
Transcript of Quantitative Methods - Probabilities and the Normal Curve
THE NORMAL CURVE
Other probability options
Other probability options
Finding a z-score continued
Finding a z-score
Other probability options
USING TABLE A
Using Table A - continued
Areas under the normal curve
μ = 100
σ = 10
A distribution of female IQ scores
A distribution of male IQ scores
Examples of normal curve
DEFINITION: The relative likelihood of occurrence of any given outcome or event
Always varies between 1.0 and 0
0.5 = 5 chances out of 10
0 = impossible (close to 0 = event is very unlikely)
1 = certainty (close to 1 = event is very likely)
# of times the outcome or event can occur
total # of times any outcome or event can occur
EXAMPLE: What is the probability of picking the ace of spades in a full deck of cards?
P = 1/52
P = 0.019
The Probability Line
Can be displayed like frequency distributions (i.e. in histograms, etc.)
Like frequency distributions, probability distributions can take a variety of shapes
We can calculate the Mean of a probability distribution
Tells us the “most expected value” in the long run
Example: The mean number of heads in 2 coin flips is 1 because the “most expected” outcome is to get 1 head and 1 tail.
Mean in a probability distribution =
Variance in a probability distribution =
σ 2 (sigma squared)
Standard Deviation in a probability distribution =
the normal curve
Interpreting standard deviation
Making statements about probability
What Is Normal?
Quite a lot!
IQ Test scores
For many things, the bigger the sample, the more normal the distribution
1, 2, 3, rule
68.26% of area between 1 and -1
95.44% of area between 2 and -2
99.74% of area between 3 and -3
34.13% of IQ scores fall between 100 and what score (below the mean)?
What is the bracket of IQ scores for 99.74% of women?
What is the bracket of IQ scores for 68.26% of men?
μ = 100
σ = 15
Now we know about areas under the normal curve
(mean, standard deviation, percentages of cases that lie within units of standard deviation…)
How do we determine the percentage of cases that lie between units of standard deviation?
FOR EXAMPLE -- what percentage of cases fall between the mean and 1.25 standard deviations above the mean?
That's where TABLE A comes in.
In this case:
z = 1.25
Area between mean and z = 39.44%
Area beyond z = 10.56%
Table A (pages 361-364) shows us:
(a) “z” -- stands for units of standard deviation
(b) “Area between mean and z” – in percentages (for only one side of the curve)
(c) “Area beyond z” – in percentages (for only one side of the curve)
Have you heard of these guys?
AKA “units of standard deviation”
AKA “sigma distances”
AKA “standard score”
How do we determine the z-score for a result within our data set?
EXAMPLE: Catherine Humes collected data about the number of jelly beans people ate in one sitting and found that:
the mean # was 13
the standard deviation was 2.
How can I find the z-score for 16 jelly beans?
μ = 13
σ = 2
1. Find the deviation from the mean
16 – 13 = 3
***16 is 3 units above the mean***
2. Divide this total by the standard deviation
3 / 2 = 1.5
What does this mean? Give me the numbers.
***This tells us that 16 jelly beans has a z-score of 1.5***
***AKA 16 jelly beans is 1.5 standard deviations above the mean***
TO SUM UP:
-z-scores: direction and degree that any given score deviates from the mean. (in units of standard deviation)
-z-score of +1.4 indicates that a score lies 1.4σ units above the mean
-z-score of -2.1 indicates that a score lies 2.1σ units below the mean
When we combine our understanding of the normal curve and z-scores (Table A)…
We can determine the probability of obtaining a score in a distribution
34.13% of scores fall between the mean and +1 σ
Convert the percentage to probability
P = 0.3413
Probability and the normal curve
Probability of getting a score between mean and +1 σ = 34.13 in 100
Finding the probability of scores on both sides of the curve
Example: Normalcurve, inc. (Founder loves normal curves and salaries are allocated and spread according to the normal curve)
What is the probability of getting a salary between $18,000 and $22,000 where the mean is $20,000 and the standard deviation is $1,500?
1) Find z-score for both options
P = 0.8164
Finding the probability of obtaining more than a single portion of the area under the curve
Example: At Normalcurve, inc., what is the probability of getting a salary either below $18,000 or above $22,000?
1) Find the z-score for both options
2) Find percentage (beyond the z) for both options using Table A
3) Convert percentage to probability for both options
4) Add probability for both options together
P = 0.1836
Finding the score value based on knowing score values for portions of the area under the normal curve
Example: At Normalcurve, inc., what is the salary level that defines the top 10.03% of earners?
1) Locate 10.03% in Table A (column c)
2) Locate z-score for 10.03% within Table A
3) Reverse our z-score formula
score – mean / standard deviation = z-score
z-score x standard deviation + mean = score
Part 1: Men’s Heights
The distribution of heights of 21 year-old men has a mean of 175 centimetres and a standard deviation of 6 cm. Using this information, answer the following questions and show your work.
1. What percentage of men are taller than 190 cm?
2. Between what heights do the middle 68.26% of men fall?
3. What is the probability of a man being shorter than 160 cm?
Part 2: Length of Pregnancies
The length of human pregnancies from conception to birth varies with a mean of 266 days and a standard deviation of 16 days. Using this information, answer the following questions and show your work.
1. Between what values do the middle 95% of all pregnancies fall?
2. How long are the longest 2.5% of pregnancies?
10% of earners = $ 21 920 or more
2) Find percentage (between the mean) for both options using Table A
3) Convert percentage to Probability for both options
4) Add Probability for both options together
Mean, median and mode coincide
43.32% between mean and z
6.68% beyond z