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STATS 100 Chapter 3 NOTES
Terms in this set (22)
standardized value (z-score)
a measure of how many standard deviations you are away from the norm (average or mean). - means below the mean, and + means above the mean.
z = (x - mean)/standard deviation
Formula for standardized value
1) Standardized values are unit free, in effect they have no units
2) A standardize value of 1.56 tells us the data value of X is 1.56 standard deviations above the mean
3) A standardize value of -0.37 tells us the data value of X is 0.37 standard deviations below the mean
4) A standardize value of 0.00 tells us the data value of X is the mean
5) The more extreme standardized value is (the further disease value is away from zero), the less likely the data value is to occur or the river that occurrence is
6) most standardized values will fall between -3.00 and +3.00
1) Standardized values are ________ free, in effect they have no ________
2) A standardized value of 1.56 tells us the data value of X is ________ standard deviations ________ the mean
3) A standardize value of -0.37 tells us the data value of X is ________ standard deviations ________ the mean
4) A standardize value of ________ tells us the data value of X is the mean
5) The more extreme the standardized value is (the ________ the value is away from zero), the ________ likely the data value is to occur or the ________ that occurrence is
6) most standardized values will fall between ________ and ________
1) shifts the entire data set by subtracting the mean from every data value in the data set. Thus, the new meaning of our standardize values is zero (z=0)
2) rescale the data set by dividing every value in the data set by the standard deviation. Thus, the new standard deviation for the standardized value is one SD for z=1
the new mean = old mean + constant, and new SD = old SD
In general, when we shift the data set (add/subtract a constant to every data value)...
the new mean = old mean x constant, and new SD = old SD x constant
In general, when we rescale the data set (multiply/divide a constant by every data value)...
The distribution is unimodal and symmetric with a low possibility of an outlier
A normal model says...
Models are theoretically excepted as being true or correct. Models are not based on any particular data set
What are theoretically expected of models?
x = variable of interest
~ = is distributed as
N = normal => unimodal and symmetric
μ = "mu" = mean of model or population mean
σ = "sigma" = standard deviation of model or population SD
Notation for normal models
The 68, 95, 99.7% rule of a Normal Model
- Approximately 68% of all data values fall within one standard deviation of the mean
- Approximately 95% of all data values fall within two standard deviation of the mean
- Approximately 99.7% of all data values fall within three three standard deviation of the mean
A Density Curve is a curve that...
...is always on or above the horizontal axis
...has an area of exactly 1 underneath it
...The median is the equal areas point, which divides the area under the curve and half
...The mean of a density curve is the balance point, at which the curve with balance it made of solid material
... if it is symmetric, the mean and the median would both lie at the center of the curve
... if it is skewed, the mean is pulled away from the median in the direction of the long tail
In a density curve...
The mean and standard deviation of the actual distribution represented by the density curve are denoted by μ ("mu") and σ ("sigma"), respectivly
While the mean and standard deviation computed from actual observations (data) are denoted as "x-bar" and "s" respectively...
Normal curves (which describe Normal distributions)
Curves that are symmetric, single kid, and bell-shaped
Causes the normal curve to move along the horizontal axis without changing it to variability
Changing the mean (μ) without changing the standard deviation (σ)...
1) In a normal curve, the standard deviation (σ) controls the VARIABILITY of a Normal curve.
2) When the standard deviation is larger, the area under the normal curve is LESS CONCENTRATED ABOUT THE MEAN.
3) The standard deviation is the distance from the CENTER to the change of curvature points on either side.
1) In a normal curve, the standard deviation (σ) controls the _______________ of a Normal curve.
2) When the standard deviation is larger, the area under the normal curve is...
3) The standard deviation is the distance from the _________________ to the change of curvature points on either side
Left (that means you need to subtract the total area, one, from the area to the left in order to get the area to the right) easyw
The SNC table only gives area to the
The same, due to symmetry
If you are given the area to the left of 1.53, the area of -1.53 would be...
You take the area to the left of the negative value in the area to the right of the positive value and subtract them
How do you find the area in the middle?
1) Turn that percentage into a decimal
2) look in the middle of the SNC table for that value, where the closest value to it
3) if there is not a specific value, take the two closest values and do the midpoint formula with them. (X1 + X2) ÷2
If you were told to find something in the nth percentile,
Carries Forward method
When you are given the X, the mean, and the standard deviation if you are told to solve for the area, use the Z score formula to solve for the Z score, which you can then use with the SNC table to find the percentage/percentile
Carries backward method
When you want just the X or percentile, and you were given a percentile, use the SNC table to figure out the Z score, then use the Z score formula to solve for X
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