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A continuous random variable has a ________ _________ if its values are spread evenly over the range of possibilities. The graph results in a ____________ shape.
- Uniform Distribution
The graph of a continuous probability distribution is called a _________ _______.
The distance along the horizontal scale of a standard normal distribution is the _________
What is the procedure for finding a z score (or nth percentile) from a known area?
- 1. Draw a bell shaped curve and identify region that corresponds to the given probability.
- 2. Locate the closest probability in the table and identify the corresponding z score.
The z score on the borderline separating the z scores that are likely to occur from those that are unlikely to occur is called the ________ __________
denotes the _________ with an area of
to its right.
We can convert nonstandard normal distribution values to standard z-scores using the formula _______ . This is rounded to ___ decimal places.
2 decimal places
What is the formula for finding x values using the z score?
The ____________ is the distribution of sample means, with all samples having the same sample size n taken from the same population.
sampling distribution of the mean.
The ______________ is the distribution of sample variances, with all samples having the same sample size n taken from the same population.
Sampling distribution of the variance.
is the _____________________
Population standard deviation
The __________________ is the distribution of sample proportions, with all samples having the same sample size n taken from the same population.
sampling distribution of the proportion
Notations for Proportions:
- p = Population proportion
- = sample proportion
Two important properties of the sampling distribution of the proportion:
- 1, The sample proportions target the value of the population proportion. (equal to the expected value)
- 2. The distribution of sample porportions tends to be a normal distribution.
What are the unbiased estimators? (4)
What are the biased estimators? (2)
- 1. Mean
- 2. Variance
- 3. Proportion
- 4. Standard deviation
What is an unbiased estimator?
They target the value of the population parameter.
What is a biased estimator?
They do NOT target the population parameter.
What are the 2 reasons for sampling with replacement?
1. when selecting a relatively small sample from a large population, it makes no significant difference whether we sample with replacement or without replacement.
2. Sampling with replacement results in independent events that are unaffected by previous outcomes, and independent events are easier to analyze and result in simpler calculations and formulas.
For a population with any distribution, n > 30, then the sample means have a distribution that can be approximated by a normal distribution with mean
and standard deviation _________.
30 and the original population has a normal distribution, then sample means have a normal distibution with mean
and standard deviation ________
30 and the original population does not have a normal distribution, then ____________
...the distribution cannot be approximated with mean
and the standard deviation
If all possible random samples of size n
are selected from a population with mean
and standard deviation
, the mean of the sample means is denoted by ______ and the standard deviation of the sample means is denoted by ______.
- sample means =
- standard deviation =
is called the
standard error of the mean
When working with an individual value from a normally distributed population use z = ?
When working with a mean for some sample (or group), use z = ?
This is called the ______________
The Central Limit Theorem
According to the ______________, for a large sample size, the sampling distribution of the sample mean is approximately normal, irrespective of the shape of the population distribution. The sample size is usually considered to be large if _______.
Central Limit Theorem
What is the Rare Event Rule for Inferential Statistics?
If, under a given assumption, the probability of a particular observed event is exceptionally small (less than 0.05), we conclude that the assumption is probably not correct.
- without replacement
- sample size n > 5% of the finite population size N (n > 0.05N)
adjust the standard deviation of sample means
by multiplying it by the ________________:
finite population correction factor:
A binomial probability distribution has what 4 requirements?
- 1. Fixed number of trials
- 2. Trials are independent
- 3. Outcome of Success or Failure
- 4. P(success) is same for each trial
What are the following notations in a Binomial probability distribution: n, x, p, q?
- n = fixed number of trials
- x = specific number successes in n trials
- p = probability of success in one of the n trials
- q = probability of failure in one of the n trials
What are the requirements for a Normal Distribution as an Approximation to the Binomial Distribution?
1. Sample is a simple random sample of size n
which the proportion of successes is p
5 and nq
1. x successes among n trials is an unusually _________ number of successes if P(x or more) is very small (such as 0.05 or less)
2. x successes among n trials is an unusually _________ number of successes if P(x or fewer) is very small (such as 0.05 or less).
What is a normal quantile plot (normal probability plot)?
What values do x and y represent?
1. A normal quantile plot is a graph of (x,y) points to determine normality.
- 2. x is the original set of data values.
- y is the corresponding z-scores that is a quartile value expected from the standard normal distribution.
How is normality determined?
- 1. Construct a histogram
- 2. Outliers - reject if more than one.
- 3. Construct a normal quantile plot
In a normal quartile plot how is normality determined?
- - pattern of points reasonably close to a straight line.
- Not Normal
- - points do not lie reasonably along a straight line.
- - show systematic pattern that is not a straight-line.
What are the steps to manually constructing a Normal Quantile Plot
- 1. Sort data
- 2. Calculate proportion for each value.
- 3. Find z scores corresponding to the cumulative left areas found in step 2.
- 4. Plot points on scatterplot
- 5. determine if normal.
What is Data Transformations?
When a distribution is not normal, it can be transformed by replacing each value of x with log(x+1), which is referred to as a lognormal distribution. Other transformations, such as
, 1/x, or
can be used.
The population proportion, p is calculated as:
- x = number of elements in the population with a specific characteristic
N= total number of elements in the population.
The sample proportion,
is calculated as:
- x = Number of elementsin the sample with a specific characteristic.
- n = total number of elements in the sample.