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Statistics Final Review
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Key Concepts:
Terms in this set (25)
Parameter
Numbers that summarize data for an entire population
Example: Average height for all men over the age of 20
Statistic
Numbers that summarize data for a sample population
Example: Average height for 45 randomly selected men over the age of 20
Weighted Mean
An average of a sample or population in which some of the data carry a higher "weight" than others
Example: Suppose that homework counts 10%, quizzes 20%, and tests 70%.
If Pat has a homework grade of 92, a quiz grade of 68, and a test grade of 81, then
Pat's overall grade = (0.10)(92) + (0.20)(68) + (0.70)(81)= 79.5
Chebyshev's Theorem
The proportion of values from a data set that will fall within k standard deviations of the mean will be at least
1 - 1/k^2
where k is (the number within)/(the standard deviation)
Use Chebyshev's theorem to find what percent of the values will fall between 123 and 179 for a data set with mean of 151 and standard deviation of 14.
151-123 = 28
179-151 = 28
--> "within number" is 28
--> k=the within number/the standard deviation=28/14=2
--> 1 - 1/(k)^2
= 1 - 1/(2)^2
= 3/4
Nominal
Nominal scales are used for labeling variables which must be mutually exclusive and have no numerical significance
Examples: Gender, Hair color, Locations, etc.
Ordinal
The order of values is whats important, but difference between each value is insignificant.
Examples:
- High school class ranking
(1st, 9th, 87th)
- Socioeconomic status (poor,
middle class, rich)
- Level of agreement (yes, no,
maybe)
- Time of day (morning,
afternoon, night)
Interval
Numerical scales in which we know not only the order, but also the exact differences between the values.
Examples:
- Temperature
- IQ (intelligence scale)
- SAT scores
- Time on a clock
Ratio
Measurement scale that tells us about the order, the exact value between units, AND have absolute zero.
Examples:
- Age
- Weight
- Height
- Ruler measurements
- Years of education
- Number of children
Mutually Exclusive
Two events that cannot occur simultaneously. For example, it is not possible to roll a five and a three on a dice at the same time.
Independent events
Two events are said to be independent of each other, meaning that the probability that one event occurs in no way affects the probability of the other event occurring.
Example:
- rolling a die and flipping a
coin
Conditional Probability
The probability of an event occurring given that another event has already occurred.
P(B I A) = P(A and B) / P(A)
Probability Distribution
A probability distribution is a table or an equation that links each outcome of a statistical experiment with its probability of occurrence.
Example:
Number of heads Probability
0 0.25
1 0.50
2 0.25
Binomial Distribution
A specific probability distribution used to model the probability of obtaining one of two outcomes. Includes a certain number of times (k), out of a fixed number of trials (N) of a discrete random event.
Probability of success = p Probability of failure = 1-p
Rules:
- Mutually exclusive
- fixed number of trials
- Each trial is independent
- Probability for success is
fixed
Normal Distribution (normalcdf)
This function returns the cumulative probability from zero up to some input value of the random variable x. Technically, it returns the percentage of area under a continuous distribution curve from negative infinity to the x.
Syntax:
Normalcdf (lower bound, upper bound, mean, standard deviation)
invNorm
This function returns the x-value given the probability region to the left of the x-value. (0<area<1 must be true)
Finds the precise value at a given p
Central Limit Theorem
Given a sufficiently large sample size from a population, the mean of all samples from the same population will be approximately equal to the mean of the population.
- If n>30, then sample average=population average
Confidence Intervals
Describes the uncertainty of a sampling method. Most often will be 90%, 95%, or 99% confidence levels.
1) Find Margin of Error
= Critical Value * Std Dev
2) Specify the interval
= Critical Value +/- Margin of
Error
Confidence Interval Example:
Suppose we want to estimate the average weight of an adult male in Dekalb County, Georgia. We draw a random sample of 1,000 men from a population of 1,000,000 men and weigh them. We find that the average man in our sample weighs 180 pounds, and the standard deviation of the sample is 30 pounds. What is the 95% confidence interval?
1) Find Standard Error
= std dev /sqrt(n)
= 30/sqrt(1000)
= 0.95
2) Critical value for 95% conf.
= 1.96
3) Margin of Error
= critical value * std error
= 1.96 * 0.95 = 1.86
Therefore, this 95% confidence interval is equal to 180 +/- 1.86
= (178.14, 181.86)
Z test
Used when n>30 or when the population standard deviation (sigma) is known
T test
Used when n<30 or when the population standard deviation (sigma) is unknown
1 Prop Z-test
Performs a Z test to compare a population proportion to a hypothesis value.
p0 = Null Hypothesis
p^ = test statistic
x = number of successes
(n-x) = number of failures
- Both x and (n-x) must be
greater than 5
Type I Error
The incorrect rejection of a true null hypothesis (a "false positive")
Type II Error
The failure to a reject a false null hypothesis (a "false negative")
Correlation Coefficients
measure the strength of association between two variables (typically the linear association between variables)
- always between -1 and 1
- greater the absolute value of a
correlation, the stronger the
relationship
- Positive correlation means that
as one variable gets bigger, the
other variable gets bigger
- Negative correlation means that
as one variable gets bigger, the
other variable gets smaller
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