BUSS1020 Chapter Notes - Chapter 6: Standard Score, Sampling Distribution, Standard Deviation
28 May 2018
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CHAPTER 6: THE NORMAL DISTRIBUTION AND OTHER CONTINUOUS DISTRIBUTIONS
• Continuous RV: can assume any value on a continuum (uncountable number of variables)
THE NORMAL DISTRIBUTION: aka Gaussian distribution
• Properties:
o Bell shaped curve
o Symmetric à skewness = 0 à mean = median = mode
o Infinite range
• The Standardized Normal Distribution: aka Z distribution
o Any normal distribution can be transformed into the standardized normal distribution Z by:
§
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o Probability is measured by the area under the curve
o Convert everything to < so we can apply the table
THE UNIFORM DISTRIBUTION: aka rectangular distribution
• Has equal density for all possible outcomes of the random variable à range a to b
o
• Probability:
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=+><+
o Mean:
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B
+ standard deviation:
C "
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:
E
FB
THE EXPONENTIAL DISTRIBUTION:
• Used to model the length of time between two occurrences of an event e.g. time between arrivals
o Positive valued, right skewed à range = 0 to positive infinity
• Probability:
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:
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,
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o Mean = standard deviation = 1/
N
+
CHAPTER 7: SAMPLING DISTRIBUTIONS
• We are interested in the distribution of the sample mean/proportion to make inferences about the population
based on the sample data
• Central Limit Theorem (CLT):
o Assumptions:
§ The sample is random
§ The sample size is large enough à n ≥ 30 OR:
§ The proportion of the sample is high enough à
OP
≥ 5
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(1 −
P
) ≥ 5
SAMPLING DISTRIBUTION OF THE MEAN:
• Unbiased property: mean of all possible sample means = population mean, µ
• Standard error of the mean: measure of the variability in mean from sample to sample
o
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R+
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S
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à Note: SE decreases as sample size (n) increases
• Normal Populations:
o If pop. is normally distributed the sampling distribution of the mean is also normally
distributed, regardless of sample size à
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R+
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o Z-value:
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U
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C
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- a = minimum value of X
- b = maximum value of X
Mean µ = 0
SD C = 1
X = arrival time
e = natural log
x = continuous variable where 0 < x < infinity
N = 1/µ = pop mean no. arrivals per unit time
V = sample mean
µ = population mean
σ = pop standard
deviation
Document Summary
Chapter 6: the normal distribution and other continuous distributions. Continuous rv: can assume any value on a continuum (uncountable number of variables) Properties: bell shaped curve, symmetric skewness = 0 mean = median = mode. The standardized normal distribution: aka z distribution: any normal distribution can be transformed into the standardized normal distribution z by: Uvs+csoc cnx. sv. p+ = 5, : probability is measured by the area under the curve, convert everything to < so we can apply the table. The uniform distribution: aka rectangular distribution: has equal density for all possible outcomes of the random variable range a to b a = minimum value of x. = maximum value of x: probability: i(% ) = , mean: > = sam. ) + standard deviation: e = 1(m,s)b: used to model the length of time between two occurrences of an event e. g. time between arrivals, positive valued, right skewed range = 0 to positive infinity.
