MKTG-310 Lecture Notes - Lecture 22: Confidence Interval, Test Statistic, Dependent And Independent Variables
Statistic review
Describing data with numbers
• Graphs are needed to show the distribution of the data
• Numerical summaries are needed to get the estimates of the population location and
spread
• Measures of location, or central tendency include
o Mean
o Median
o Mode
• Measures of spread include
o Range
o Standard deviation
o Variance
Measures of central tendency
• The mean is the average of all the values
o To calculate add all the observations and divide that total by the number of
observations
• The median is the middle value, 50% of the data is larger than this number and 50% of
the data is smaller than this number
o To calculate, order the data from smallest to largest. If you have an odd number
observation, select the middle number as the median. If you have an even number
of observations, take the average of the middle 2 numbers
• The mode is the most frequently occurring value or values
o To calculate, tally all of the values and see which one or ones occur the most
often
o You can have no mode or more than 1 mode for a set of data
• When you have data that are normally distributed, the mean, median and model are all
very close to the same value
Measures of spread
• The range is the difference between the largest and the smallest values
o To calculate subtract the minimum value from the maximum value
o Range depends on sample size. An order of magnitude estimates of the standard
deviation can be derived from the range. If the range is of 5 numbers, the order of
magnitude estimate of the standard deviation is the range/ 2.5. if the range is
based on ten values, the order of magnitude estimate of the standard deviation is
the range/ 3. If the range is based on 25 values the order of magnitude estimate of
the standard deviation is the range/4.
• The variance is like the average squared distance each observation is from the mean
o To calculate you first need to find the mean of the data. Then you need to take the
difference between each observation and the average squared, add those up and
divide by the number observations minus 1.
• The standard deviation is the square root of the variance
o To calculate take the square root of the variance
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o The standard deviation is use because it has the same units of measure as the
observation
The normal distribution
• Measurements from the population are not all exactly the same. They vary but together
make up a distribution of values
• Measurements often follow the normal distribution
o Roughly 68% of the data falls within +/- 1 standard error of the mean
o Roughly 95% of the data falls within +/- 2 standard errors of the mean
o 99.73% of the data falls within +/- 3 standard errors of the mean
The sampling distribution of the mean
• In the previous slide we state that xx% of the distribution is within +/1 y standard errors
of the mean
• If we are working with individual data points, the standard error is the same as the
standard deviation
• If we are working with means of n data points, the standard error of the mean is equal to
the standard deviation of the individuals Square root (n)
Confidence intervals
• If we are sampling from a normal distribution, then we expect about 95% of your samples
to be within +/- 2 standard errors of the true mean (see black| in figure at right)
• If we take our sample mean, and add and subtract about 2 standard errors from that value
(the length of the red double-sided arrows in the figure at the right), the interval we make
should contain the true population value for 95 out of 100 intervals
• A sample more than 2 standard errors from the population mean will not create an
interval that contains the population mean
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