M. Drew LaMar
January 30, 2019
“Statistics are used much like a drunk uses a lamppost: for support, not illumination.”
- Andrew Lang
Definition: The
sampling distribution represents the distribution of the point estimatesbased on samples of a fixed size from a certain population. It is useful to think of a particular point estimate as being drawn from such a distribution. Understanding the concept of a sampling distribution is central to understanding statistical inference.
Definition: The standard deviation associated with an estimate is called the
standard error . It describes the typical error or uncertainty associated with the estimate.
The standard error is also the standard deviation of the sampling distribution.
http://www.zoology.ubc.ca/~whitlock/kingfisher/SamplingNormal.htm
Definition: The standard error represents the standard deviation associated with the estimate, and roughly 95% of the time the estimate will be within 2 standard errors of the parameter.
An approximate 95% confidence interval for a point estimate is given by \[ \textrm{point estimate} \pm 1.96\times SE \]
Note: For a yuge number of computed 95% confidence intervals, the population parameter will be contained in 95% of the confidence intervals.
If a sample consists of at least 30 independent observations and the data are not strongly skewed, then the sampling distribution for the mean is well approximated by a normal model.
Definition:
Hypothesis testing compares data to what we would expect to see if a specific null hypothesis were true. If the data are too unusual, compared to what we would expect to see if the null hypothesis were true, then the null hypothesis is rejected.
Definition: A
null hypothesis is a specific statement about a population parameter made for the purpose of argument.
Definition: The
alternative hypothesis includes all other feasible values for the population parameter besides the value stated in the null hypothesis.
Can parents distinguish their own children by smell alone? To investigate, Porter and Moore (1981) gave new T-shirts to children of nine mothers. Each child wore his or her shirt to bed for three consecutive nights. During the day, from waking until bedtime, the shirts were kept in individually sealed plastic bags. No scented soaps or perfumes were used during the study. Each mother was then given the shirt of her child and that of another, randomly chosen child and asked to identify her own by smell.
Discuss: What is the
null hypothesis ?alternative hypothesis ?
Can parents distinguish their own children by smell alone? To investigate, Porter and Moore (1981) gave new T-shirts to children of nine mothers. Each child wore his or her shirt to bed for three consecutive nights. During the day, from waking until bedtime, the shirts were kept in individually sealed plastic bags. No scented soaps or perfumes were used during the study. Each mother was then given the shirt of her child and that of another, randomly chosen child and asked to identify her own by smell.
Discuss: What is the
null hypothesis ?alternative hypothesis ?
Answer: With \( p \) the probability of choosing correctly,
\[ H_{0}: \ p = 0.5 \] \[ H_{A}: \ p \neq 0.5 \]