Question : A major disadvantage of the median, when compared to the mean, is that
a. its location cannot be calculated algebraically.
b. it is disproportionately affected by outliers.
c. it is not appropriate for use with nominal data.
d. it is not appropriate for use with ordinal data.
e. it is less stable than the mean from sample to sample.
Answer : B.
Question : This question has to do with central limit theorem.
Suppose a random number generator generates real numbers with uniform frequency between 1 and 5. If we randomly sample 75 random numbers, determine the following.
b. Xbar ~ 'blank' X~ 'blank'
c. draw and label both distributions from b.
(you don't really have to answer this if there is no way to draw it online)
d. the probability that the sample mean is no more than 2.5
e. find the probability that any given rand..
Answer : Let X1, X2, ... , Xn be a simple random sample from a population with mean and variance .
Let Xbar be the sample mean = 1/n * Xi
Let Sn be the sum of sample observations: Sn = Xi
then, if n is sufficiently large:
Xbar has the normal distribution with mean and variance / n
Xbar ~ Normal( , / n)
Sn has the normal distribution with mean n and variance n
Sn ~ Normal(n , n )
The great thing is that it does not matter what the under lying distribution is, the central limit theorem holds. It was proven by Markov using continuing fractions.
if the sample comes from a uniform distribution the sufficient sample size is as small as 12
if the sample comes from an exponential distribution the sufficient sample size could be several hundred to several thousand.
if the data comes from a normal distribution to start with then any sample size is sufficient.
for n < 30, if the sample is from a normal distribution we use the Student t stat..
Question : Determine whether the hypothesis test involves a sampling distribution of means that is a normal distribution, Student t
distribution, or neither.
Claim: = 118. Sample data: n = 17, x = 101, s = 15.4. The sample data appear to come from a
normally distributed population with unknown and .
Answer : Small Sample Hypothesis Test for mean:
In order for this test to be valid the data must come from a normal population. If this is not the case then this test is not valid and other methods, such as a randomization test or permutation test should be used.
Assuming the normality assumption is valid to test the null hypothesis
Find the test statistic t = (xbar - ) / (sx / (n))
where xbar is the sample average
sx is the sample standard deviation, if you know the population standard deviation, , then replace sx with in the equation for the test statistic.
n is the sample size
and t follows the Student t distribution with n - 1 degrees of freedom. We use the Student t distribution to account for the uncertainty in the estimate of the variance.
As the degrees of freedom approach infinity the Student t converges in probability to the Standard Normal. In most cases the values of the percentiles of the Stude..