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Define discrete.
A discontinuous [finite] number of outcomes.
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Steps for Discrete Probability Distribution:
- 1) Define a variable.
- 2) Create a discrete probability distribution
- 3) Determine if the distribution is valid
- 4) Compute the mean or expected value of x
- 5) Compute the variance and standard deviation
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Define binomial distribution.
The discrete probability distribution of the number of successes in a sequence of n independent yes/no experiments, each of which yields success with probability p.
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What would x in a discrete probability distribution problem be referred to as?
x is called the discrete random variable.
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Discrete Probability Distribution
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What are some properties that make a distribution valid?
- 1) Each probability must be a fraction [0 ≤ probability ≤ 1]
- 2) The sum of all probabilities equal up to 1 [∑ probabilities = 1]
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The following formula refers to the whole population if the parameter is considered to be discrete/finite (it is used to compute the mean or expected value of x):
μ = ∑x p(x)
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The following formula is used to compute the variance and/or standard deviation:
σ² = ∑x² p(x)-μ²
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In discrete probability distributions, each event should be considered to be ______________________ of another.
independent
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Explain what is being done with this formula:
P(GGG)= p(G) p(G) p(G)
The probability at each stage is being multiplied.
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What are the four properties of a binomial experiment?
- 1) The experiment has been repeated more than once
- 2) There must be only two possible outcomes
- 3) Repetitions of the experiment must be independent of each other
- 4) Probabilities must remain constant from trial to trial
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To compute the probabilities of a binomial experiment, you must use the binomial formula:
P(x=r)=C(n,r)·pr·(1-p)(n-r)
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In the formula: P(x=r)=C(n,r)·pr·(1-p)(n-r), what are the p and n values?
The n [number of trials] and p [probability of success] are called the parameters of the Binomial Distribution and must always be specified in a problem in order to work out the problem.[Note: The r can be any number greater than or equal to 0.]
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What does the C(n,r) represent?
C(n,r) represents the number of ways to arrange r successes out of n trials.
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The combination formula:
- C(n,r)= n!/[r!(n-r)!]
- The (n,r) can be thought of as a coordinate plane pair (x,y). [Note: n!=n(n-1)(n-2)(n-3)...3·2·1; i.e. 0!=1, 1!=1, 2!=2·1=2, 3!=3·2·1=6, 4!=4·3·2·1=24]
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What are the two shortcut formulas for a binomial experiment?
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Which formula would be best to solve for the binomial number?
μ=np
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Which formula would be best to solve for the variance?
σ²=np(1-p)
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What determines whether an answer is usual?
In order to be considered a usual amount, a number must fall between two standard deviations of a distribution’s mean [μ ± 2σ].
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What is continuous probability distribution?
Continuous probability distribution is when a probability distribution has a cumulative distribution function that is continuous. That means, the variable x has no probability of attaining the value r, therefore its probability will have a zero value [P(x=r)=0]. The x in this case is referred to as a continuous random variable.
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In reference to a graph, what property of that graph is equivalent to its probability?
A graph’s area/density/thickness can be used to determine the probability of a graph’s component(s).
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While a 1)_____________________ graph can be used as a continuous probability distribution, the 2)__________________________ graph is more commonly used by statisticians.
- 1) uniform
- 2) bell-shaped or normal distribution or Gaussian distribution or bell curve or symmetric shape [might want to know these various names]
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Bell-shaped curve or normal distribution or Gaussian distribution or bell curve or symmetric shaped curve
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What determines whether an amount is usual in a continuous probability distribution?
The amount must be greater than 5%.
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This approach is the one approach used in determining the area of the bell-shaped curve in a calculus-based course.
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The normal distribution function
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Which continuous probability distribution graph does Z distribution refer to?
It refers to the standard normal distribution.
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What do the numbers found in the Z-score table refer to?
The numbers found in the Z-score table refer to the probabilities to the left of a number found on a normal distribution graph.
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In the continuous probability distribution, what two properties are relevant?
- 1) The sum of all probabilities must equal 1 [∑ probabilities=1]
- 2) Zero is always in the middle of a standard normal distribution
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What formula should be used when converting a scale to the Z-scale?
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x has a binomial distribution with the parameters 1)_________ and 2)__________.
- 1) n
- 2) p
- [Note: It is written in (n,p) notation.]
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The y has a binomial distribution that differs in its parameters from the x distribution. What are its parameters? (Write it in the correct notation.)
n(μ,σ²)
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The variables/parameters used in statistics are referred to as ________________.
unbiased estimators
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What two functions best normalize a distribution graph?
- 1) taking the square root of x
- 2) log x
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What are the six graphs viewed while assessing normality?
- 1) Histogram
- 2) Stem and leaf
- 3) P-P plots
- 4) Q-Q plots
- 5) Skewness
- 6) Kurtosis
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Of the six graphs of assessing normality, which two are used for standard normal distributions?
The P-P plots and Q-Q plots are used when assessing the normality of a standard normal distribution plot.
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Of the six graphs viewed while assessing normality, which of the six is used to assess the normality of a numerical quantitative distribution?
Kurtosis is used when assessing the normality of a numerical quantitative distribution.
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What is the Central Limit Theorem?
The central limit theorem states that the mean of an independent random variable will be normally distributed [meaning that it will correlate with the standard normal distribution].
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Standard deviation _______________ with larger population samples.
decreases
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If x has a non-normal distribution, then the rule n___30 must be applied.
≥
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The formula used to compute probabilities about the mean of x:
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