03/03/2021
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The following topics will be covered in this lecture:
Our goal in this course is to use statistics from a small, representative sample to say something general about the larger, unobservable population or phenomena.
Recall, the measures of the population are what we referred to as parameters.
Parameters are generally unknown and unknowable.
Random variables and probability distributions give us the model for estimating population parameters.
Generally, we will have to be satisfied with estimates of the parameters that are uncertain, but also include measures of “how uncertain”.
Courtesy of M. W. Toews CC via Wikimedia Commons.
The proportion (or fraction) of any set of data lying within \( K \) standard deviations of of the mean is always at least \( 1-\frac{1}{K^2} \) where \( K>1 \).
The word variance also has a specific meaning in statistics and is another tool for describing the variation / dispersion / spread of the data.
Suppose that the data has a population standard deviation of \( \sigma \) and a sample standard deviation of \( s \).
Then, the data has a population variance of \( \sigma^2 \).
Likewise, the data has a sample variance of \( s^2 \).
Therefore, for either a population parameter or a sample statistic, the variance is the square of the standard deviation.
For example, measuring the heights of students in inches, the standard deviation is in the units inches.
Outcome | Observed value for \( X=x \) | Probability |
---|---|---|
\( \{H,H\} \) | \( x=2 \) | \( f(x)=\frac{1}{4} \) |
\( \{H,T\}, \{T,H\} \) | \( x=1 \) | \( f(x)=\frac{2}{4} \) |
\( \{T,T\} \) | \( x=0 \) | \( f(x)=\frac{1}{4} \) |
the probability of three independent successful trials, times the probability of two independent failure trials, times all possible ways we can have exactly \( 3 \) successful trials out of five.
par(mai=c(1.5,1.5,.5,.5), mgp=c(3,0,0))
plot(c(0:5), dbinom(c(0:5), size=5, prob=0.85), ylab="Probability", xlab="Number of Successses", cex=3, cex.lab=3, cex.axis=2.5, main="Probability mass function", cex.main=3)
In the above, the dbinom
is the function for the probability mass function for the binomial distribution.
We are setting a range of values to plot c(0:5)
and the size of the trial size=5
and the probability of succcess prob=0.85
.
Courtesy of Ania Panorska CC