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Terms in this set (16)
basic properties of a density curve
1. A density curve is always on or above the horizontal axis.
2. The total area under a curve equals 1.
Variables and their density curve
For a variable with a density curve, the percentage of all possible observations of the variable that lie within any specified range equals the corresponding area under the curve, expressed as a percentage.
Normally distributed variable
A variable is said to be a normally distributed variable or to have a normal distribution if its distribution has the shape of a normal curve.
Normally distributed variables and normal curve areas
For a normally distributed variable, the percentage of all possible observations that lie within any specified range equals the corresponding area under its associated normal curve, expressed as a percentage. This result holds approximately for a variable that is approximately normally distributed.
Standard Normal distributions; standard normal curve
A normally distributed variable having mean 0 and standard deviation 1 is said to have the standard normal distribution. Its associated normal curve is called the standard normal curve.
Standardized normally distributed variable.
z transform by minusing the mew and dividing by sigma
Basic properties of the standard normal curve
1. The total area under the standard normal curve is 1.
2. The standard normal curve extends indefinitely in both directions, approaching, but never touching the horizon axis as it does so.
3. The standard normal curve is symmetric about 0; that is, the part of the curve to the left of the dashed line is the mirror image of the part of the curve to the right.
4. Almost all the area under the standard normal curve lies between -3 and +3
The Z notation
the symbol z subsigma is used to denote the z-score that has an area of alpha to it's right under the standard normal curve
1. 68.26% of all the possible observations lie within one standard deviation to either side of the mean, that is, between mew-sigma and mew +sigma.
2. 95.44% of all possible observations lie between two standard deviations to either side of the mean, that is, between mew-2sigma and mew +2sigma.
3. 99.74% of all possible observations lie between 3 standard deviations to either side of the mean, that is, between mew-3sigma and mew +3sigma.
is the error resulting from using a sample to estimate a population characteristic.
The sampling distribution of the sample mean
For a variable x and a given sample size, the distribution of the variable x bar is called the sampling distribution of the sample mean.
Sample size and sampling error
The larger the sample size, the smaller the sampling error tends to be in estimating a population mean, mew, by a sample mean, x bar.
Mean of the sample mean
For samples of size n, the mean of the variable x bar equals the mean of the variable under consideration. mew x bar = mew
Standard deviation of the sample mean
For samples of size n, the standard deviation of the variable x bar equals the standard deviation of the variable under consideration divided by the square root of the sample size. sigma subx bar= sigma/sqrt n
Sampling distribution of the sample mean for a normally distributed variable
Suppose that a variable x of a population is normally distributed with mean mew and standard deviation sigma. Then, for sample of size n, the variable x bar is also normally distributed and has mean mew and standard deviation sigma/sqrt n
Central limit Theorem
For a relatively large sample size, the variable x bar is approximately normally distributed, regardless of the distribution of the variable under consideration. The approximation becomes better with increasing sample size.
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