Week2| Population parameters Flashcards
There are four properties that make point estimators easier to work with and are possessed by good point estimators. Name them (HINT BLUE)
- If Beta hat is said to be a linear estimator of Beta if it is a linear function of the sample observations.
For example the sample mean X-bar is a linear estimator of the population mean mu. However, the sample variance is a quadratic function of the X-sample observations so it is a non linear estimator of population variance - Beta hat is said to be an unbiased estimator of Beta if E(Betahat)= Beta
For example, if the unexpected value of Beta hat is equal to Beta and thus sampling distribution of beta-hat is centered around beta
Otherwise, Beta is referred to as a biased estimator and
(E(Betahat) - Beta) doesn’t equal to zero
The sample mean is an unbiased estimator of the population mean because E(Xbar) = mu
Similarly, the sample variance is an unbiased estimator of the population variance
The more beta hat is centred around beta the more accurate of an estimator it will be
- Beta hat is an efficient estimator of Beta within some well-defined class of estimators. If its variance is smaller or at least not greater than that of any other estimator of Beta in the same class of estimators
(Only used to help)
Beta 3 hat and beta 4 hat are unbiased estimators of beta. However, Beta 3 hat has a smaller variance than the sampling distribution of beta-4 hat
Beta 3 will be a more accurate estimate of Beta than beta 4 hat - Beta hat is called a consistent estimator of Beta if its sampling distribution collapses in to a vertical straight line at the point Beta when the sample size n goes to infinity
The sampling distributions are all centered around Beta and as the sample size increases they become narrower
-> granted that this is true for larger sample sizes as well; Beta -hat is a consistent estimator of Beta
variance needs to go to zero for increasing n.
What do parametric procedures involve?
What are the requirements for a parametric test
2 things:
a) concerned with population parameters and
b) based on certain assumptions about the sampled population or about the sampling distribution of some point estimator
Requirements:
i) The sample has been randomly selected
<- otherwise it might not represent the population accurately
ii) The variable of interest is quantitative
iii) …and is measured on an interval or a ratio scale
<- otherwise, the population mean would not exist f the central location could be measured only with the mode and the median (if the measurement scale is at least ordinal)
iv) The population standard deviation is unknown but the population is normally distributed at least approximately
What are non-parametric tests?
What are their requirements?
Non-parametric tests are procedures that are wither not concerned with some population parameter or are based on relatively weaker assumptions than their parametric counterparts and hence require less information about the sampled population
Note: Non parametric techniques are sometimes referred to as distribution free procedures. This is a bit deceptive as they also rely on some, though fewer and less stringent assumptions about the sampled population
What is the crucial assumption behind most parametric procedures? How does this apply to the parametric tests? (Z and t test )
Normality
In the case of testing a population mean with a parametric test, either the standard deviation should be known and the sample mean should be normally distributed (Z-test)
or standard deviation is unknown but the sampled population itself should be normally distributed (t-test) sample mean is also normally distributed
In what three ways can we check up on normality?
i) Graphs (histogram and QQ plot)
QQ (quantile-quantile ) plot is a scatterplot that depicts cumulative relative frequency distribution of the sample data against some known cumulative probability distribution
If points are not on the straight line then not normal
Histogram: if skewed then not normal
ii) Quantifying normality with numerical descriptive measures:
mean, median, kurtosis and skewness
1. Mean & median If modemean (negatively skewed) left skewed
2. Skewness A descriptor of the shape of a distribution and it is concerned with the asymmetry of a distribution around its mean SK=0 (normal) SK>0 (right skewed) SK<0 (left skewed)
- Kurtosis (K) another descriptor of the shape of a (unimodal) distribution. It is about the tails of a distribution (outliers) relative to normal distribution
leptokurtic: long tails, has more outliers
Platykurtic: short tail, fewer outliers
In terms of parametric vs non parametric procedures the real issue is whether the distribution is normal or not K=3 for normal distribution K>3 for leptokurtic K<3 platykurtic K-3 excess kurtosis
iii) There are several statistical tests for normality
H0: data comes from a normally distributed population
HA: the data comes from a non-normally distributed population
We only use the SW test because it is easy to implement in R and compares favourably to other tests for normality at the limited sample size
What are the two short comings of the shapiro wilk test?
i) At small sample sizes (n<20), when the normality assumption can be crucial, it has little power to reject H0 even if the population is indeed not normally distributed (Type II error)
ii) At large sample sizes (n>100) when the violation of normality is far less critical in practice, it tends to be too sensitive to the slightest signs of non-normality in the sample and often rejects H0 even if it is actually true (Type I error)
hat are the descriptive statistics obtained from R from performing the SW test
i. The sample mean (12.175) is bigger than the sample median (10.500), so the sample of diff is skewed to the right (non-normal). (mean in R)
ii. SK-hat= 0.556 is positive, so the sample of diff is skewed to the right. SK-hat divided by twice of the standard error is skew.2SE = 1.151 >1 so the distribution of diff is unlikely normal (skewness)
iii. The estimate of excess kurtosis is K-hat– 3 = -0.548. It is negative, so the sample of diff is platykurtic. However, the absolute value of K-hat– 3 divided by twice of the standard error is |kurt.2SE| = 0.573 < 1, so the distribution of diff might be normal.(kurt 2se)
iv. The reported p-value of the SW test is normtest.p= 0.001 < 0.05, thus
normality is rejected at the 5% level. (normtest.p)
What are the benefits of the mean and the median?
Mean is a comprehensive measure because it is computed from all available data points, while the median is based on at most 2 data points
The mean is used far more extensively in inferential statistics than the median
Median advantage:
Not affected by outliers
Exists even if measurement scale is ordinal
A hypothesis about the central location of a quantitative population is
usually best tested with a ttest for the population mean ().
However, when the mean does not exists, or it is not an ideal measure
of central location due to the presence of outliers, or when the ttest is
inappropriate because the normality assumption is clearly violated,
instead of this parametric procedure one should rely on some
nonparametric alternative for testing the central location of a population.
What are the requirements for one sample sign test
Sign test for the median (n):
This procedure assumes:
i. The data is a random sample of independent observations
ii. Variable of interest is qualitative or quantitative
iii. Measurement scale is at least ordinal
But, the sign test does not require any assumption about the distribution of the sampled population
H0: n = n0 vs HA: n (not equal to) n0 , nn0
S = t-stat
S~B(n,0.5) -> E(S) = 0.5n, Var (S)= 0.25n
For sufficiently large n(np= nq= 0.5n(greater than or equal to) 5, so n(greater than or equal to) 10), this binomial
distribution (B) can be approximated with a normal distribution (N),
Reject H0 if:
right-tail test: pR = P(S (greater than or equal to) S+) is small
left-tail test: pL = P(S (greater than or equal to) S+ is small
two tail test: 2xmin(pR,pL) is small
What is the wilcoxon sign test and what are the requirement?
The Wilcoxo signed ranks test for the median (n) also known as the Wilcoxon signed rank sum test is based entirely on the signs of deviations from n0
<- The wilcoxon signed ranks test is a more sensitive and potentially ore powerful alternative because it takes the magnitudes of these deviations as well into consideration
requirements:
i) The data is a random sample of independent observations
ii) The variable of interest is quantitative and continuous
iii) The measurement scale is interval or ratio
iv) The distribution of the sampled population is ymmetric (mu = median)
Test statistic is: T=T+
0T, U, alpha
ii) left tail test: T< TL, alpha
iii) 2 tail test: T> T U, alpha/2 or T 30) the sampling distribution of T can be approximated with a normal distribution
Namely
T~N(muT, standard deviations T) with
muT = n(n+1)/4, (standard deviation T)^2= n(n+1)(2n+1)/24
When do we reject the null
when p-valule>0.05 we maintain
when p-value<0.05 we reject
Similarly to the Wilcoxon signed ranks test, many other nonparametric
tests assume that the underlying variable of interest is continuous.
This assumption is primarily required to exclude the possibility of ties and
it is necessary for exact hypothesis tests.
Still, these procedures are often used in practice even if the variable of
interest is discrete, or is reported on a discrete scale, but
i. there are a large number of different values,
or ii. when the sample size is large enough to approximate the discrete
sampling distribution of the test statistic with some continuous
probability distribution.
Because of this assumption, these tests are most appropriate for use on
data reported on an interval or ratio scale. Yet, they are occasionally
used on ordinal data as well.
