Download math1005 statistics

MATH1005 STATISTICS
[email protected]
http://mahritaharahap.wordpress.com/teaching-areas
Tutorial 10: Test for Means
1
In statistics we usually want to statistically analyse a population but collecting data
for the whole population is usually impractical, expensive and unavailable. That is
why we collect samples from the population (sampling) and make inferences about
the population parameters using the statistics of the sample (inferencing) with some
level of accuracy (confidence level).
Statistical inference is the process of
drawing conclusions about the entire
population based on information in a
sample by:
• constructing confidence intervals on
population parameters
• or by setting up a hypothesis test on a
population parameter
General: Hypothesis Testing
 We use hypothesis testing to infer conclusions about the population parameters based on
1.
2.
3.
4.
5.
analysing the statistics of the sample. In statistics, a hypothesis is a statement about a population
parameter.
Hypothesis:
The null hypothesis, denoted H0 is a statement or claim about a population parameter that is
initially assumed to be true. Is always an equality. The null hypothesis must specify that the
population parameter is equal to a single value.
The alternative hypothesis, denoted by H1 is the competing claim. What are we trying to prove.
Claim we seek evidence for. (Eg. H1: population parameter ≠ or < or > hypothesised null parameter)
Assumptions: A hypothesis test is invalid if the assumptions are not satisfied.
Test Statistic: a measure of compatibility between the statement in the null hypothesis and
the sample data obtained. It is a random variable consisting of a function of the observed
values, with a distribution depending on the unknown parameter.
P-Value: is the probability of obtaining a test statistic more extreme than the observed sample
value given the null hypothesis is true.
Conclusion: Compare the p-value with the level of significance α. If the test statistic falls in
the rejection region, p-value is small , so we reject H0 and conclude that we have enough
evidence is against H0.
If the test statistic falls in non-rejection region, p-value is large, so we do not reject H0 and
conclude that we do not have enough evidence to support H0.
Make your conclusion in context of the problem.
If H1: µ < µo
If H1: µ > µo
If H1: µ ≠ µo
P-value=P(Z<τo)
P-value=P(Z>τo)=1-P(Z<τo)
P-value=2*P(Z>|τo|)=2*(1-P(Z<|τo|))
Or
Or
Or
P-value=P(tn-1<τo)
P-value=P(tn-1>τo)=1-P(tn-1<τo) P-value=2*P(tn-1>|τo|)=2*(1-P(tn-1<|τo|))
On R use pnorm for Z test and pt for t-Test.
Both pnorm and pt give you lower tail values.
To find p-value manually, note that the Z tables give you lower tail values
and the t table give you upper tail values
Hypothesis:
H0: µ = 1800
H1: µ > 1800
Assumption:
Population variance σ2 is unknown but we know the sample variance s2
therefore we will use the t-test.
Test Statistic:
𝜏0 =
1850 − 1800
110/ 50
= 3.21~𝑡𝑛−1 = 𝑡49
P-value:
P(t49 > 3.21) = 1 –P(t49 < 3.21)=0.001
Conclusion:
Since p-value<0.05, we reject the H0, and conclude that we have enough
evidence to prove the rope strength has increased.
Note that the t table is upper tail.
Note that the t table is upper tail
and the z table lower tail.