Download College Prep Stats Chapter 5 Important Information Sheet 1

College Prep Stats
Chapter 5
Important Information Sheet
1. Random Variable: a variable (typically represented by x) that has a single numerical value, determined by chance, for each
outcome of a procedure. (Example: the number of peas with green pods among 5 offspring peas.)
2. Probability Distribution: a description that gives the probability for each value of the random variable; often expressed in the
format of a graph, table, or formula. This gives a description of what will probably happen.
3. Discrete Random Variable: either a finite number of values or countable number of values, where “countable” refers to the fact
that there might be infinitely many values, but they result from a counting process. (It cannot be a decimal.)
4. Continuous Random Variable: infinitely many values, and those values can be associated with measurements on a continuous
scale without gaps or interruptions. (If something could be a decimal, it is continuous.)
5. Requirements for a Probability Distribution:
1. ΣP(x) = 1, where x assumes all possible values. (Values such as 0.999 or 1.001 are acceptable due to roundoff error)
2. Each individual probability is a value between 0 and 1 inclusive. 0  P(x)  1, for every individual value of x.
6. Formula for Mean of a probability distribution: µ = Σ [x • P(x)]
7. Formula for Standard Deviation of a probability distribution:  =
  x
2
 P  x     2
8. Range Rule of Thumb: According to the range rule of thumb, most values should lie within 2 standard deviations of the mean.
We can therefore identify “unusual” values by determining if they lie outside these limits:
Maximum usual value = μ + 2σ
Minimum usual value = μ – 2σ
9. Using Probabilities to Determine When Results Are Unusual:
a) Unusually high: x successes among n trials is an unusually high number of successes if P(x or more) ≤ 0.05.
b) Unusually low: x successes among n trials is an unusually low number of successes if P(x or fewer) ≤ 0.05.
10. Expected Value: The expected value of a discrete random variable is denoted by E, and it represents the mean value of the
outcomes. Formula: E = Σ[x • P(x)].
11. Binomial Probability Distribution: A binomial probability distribution results from a procedure that meets all the following
requirements:
1. The procedure has a fixed number of trials.
2. The trials must be independent. (The outcome of any individual trial doesn’t affect the probabilities in the other trials.)
- If the trials are dependent but n  .05N, the events may be treated as independent.
3. Each trial must have all outcomes classified into two categories (commonly referred to as success and failure).
4. The probability of a success remains the same in all trials.
12. Notation for Binomial Probability Distributions:
n
x
p
q
P(x)
denotes the fixed number of trials.
denotes a specific number of successes in n trials, so x can be any whole number between 0 and n, inclusive
denotes the probability of success in one of the n trials.
denotes the probability of failure in one of the n trials.
denotes the probability of getting exactly x successes among the n trials.
13. Binomial Probability Formula: P  x  
n!
 p x q n  x , where
 n  x ! x !
n = number of trials
x = number of successes among n trials
p = probability of success in any one trial
q = probability of failure in any one trial (q = 1 – p)
14. Calculator Rules for Binomial Probability Distributions:
1. *Keyword “exact” (this is for equality only: example: Find the probability that if you toss a coin 10 times, it lands on tails 5 times.)
CALCULATOR SET UP: binompdf(number of trials, probability of success, number of successes) = binompdf(n, p, x)
2. *Keywords “less than or equal to” (if it says less than or fewer than you must change it to less than or equal to by doing the
following: less than 6 is the same as less than or equal to 5, fewer than 8 is the same as less than or equal to 7).
CALCULATOR SET UP: binomcdf(number of trials, probability of success, number of successes)
3. *Keywords “at least” or “greater than or equal to” (if it says greater than or more than you must change it to greater than or
equal to by doing the following: greater than 6 is the same as greater than or equal to 7, more than 8 is the same as greater than or
equal to 9).
CALCULATOR SET UP: 1 – binomcdf(number of trials, probability of success, 1 less number for success)
What you are doing here is 1 minus what you DO NOT WANT and that will give you the probability that you do want.
Example:
Calculate the probability of
tossing a coin 20 times and
getting exactly 9 heads.
Calculate the probability of
tossing a coin 20 times and
getting less than 6 heads.
Calculate the probability of
tossing a coin 32 times and
getting at least 14 heads.
Pdf or Cdf
pdf
Calculator input:
binompdf(20, 0.5, 9) = 0.160
cdf, since it says less than and our calculator
only does less than or equal to we will change
the problem to say: Calculate the probability of
tossing a coin 20 times and getting less than or
equal to 5 heads.
cdf, since it says at least and our calculator only
does less than or equal to we will have to do the
complement of at least 14 which is less than or
equal to 13. Since we are doing the complement
we must do 1 minus.
binomcdf(20, 0.5, 5) = 0.021
1 – binomcdf(32, 0.5, 13) = 0.811
15. Binomial Distribution: Formulas
Mean
µ=n•p
Variance
σ =n•p•q
Standard deviation
  n pq
2
where
n = number of fixed trials
p = probability of success in one of the n trials
q = (1 – p) = probability of failure in one of the n trials
16. Interpretation of Results
It is especially important to interpret results. The range rule of thumb suggests that values are unusual if they lie outside of these
limits:
Maximum usual values = µ + 2σ
Minimum usual values = µ – 2σ