lectures 9 & 10 Flashcards
If the absolute z is larger than the critical value:
we reject the null hypothesis.
it means that the observed test statistic is in the critical region of both tails of the distribution. In the context of hypothesis testing, this could lead to the rejection of the null hypothesis.
If the absolute z is smaller than the critical value:
we fail to reject the null hypothesis.
means that the observed test statistic is not extreme enough to fall into the critical region where you would reject the null hypothesis.
Two-tailed tests:
reject the null hypothesis if z is particularly large OR particularly low.
E.g.: H0 : µ = 100
We would reject the null hypothesis if students’ IQ is more OR less than 100.
We reject the null hypothesis if z falls in the grey zone. Either the left or the right one.
One-tailed tests
we want to test whether students’ IQ is HIGHER than the national average: H0 : µ = 100 H1 : µ > 100
We reject the null hypothesis if z falls in the grey zone.
This time, the grey zone is only on the upper-tail.
Since we still want a level of significance α = 0.05, we assign all 5% to the upper-tail.
Now we reject the null hypothesis only if we observe an outcome in the upper tail (i.e. a particularly large value)
The threshold to reject the null hypothesis in a one-tailed test is therefore lower (e.g. 1.96 vs 1.64). I.e. it is easier to reject the null hypothesis.
Thus, a two-tailed tests is recommended unless you have a very good reason not to.
p-value
the probability of observing a test statistic as extreme as, or more extreme than, the one actually observed, assuming that the null hypothesis is true. If the p-value is less than the significance level (α), you would reject the null hypothesis.
The smaller the p-value, the stronger the evidence against H0. It implies that our outcome is more “surprising”
Type I error:
True H0
Correct decision = retain H0
False alarm error = reject H0
Type II error:
False H0
Miss error = retain H0
Correct decision = reject H0
The Z test is accurate if
- the sampling distribution of the mean is normally distributed. AND
- the population standard deviation (σ) is known.
Point 2 is problematic because we usually do not know the population standard deviation (σ). All we know is the sample standard deviation (s).
The z-score depends on σ.
As we do not know σ, we could use the sample standard deviation (s) instead. However, the sample standard deviation is a biased estimator of the population standard deviation (it is likely to be lower than the population standard deviation).
Underestimating the population standard deviation (e.g. 10 instead of 15) implies underestimating the standard error & vice versa
The T Test
useful when the sample size is small
The t-distribution has a similar shape than the standard normal distribution, but is has heavier tails
The shape of the t-distribution depends on the number of degrees of freedom. The t-distribution has N-1 degrees of freedom. N is the sample size.
when are the t distribution and normal distribution the same
As the number of observations (and hence degrees of freedom) increases, By the time you have about 30+ observations, the two are virtually indistinguishable.
comparing two samples
paired samples
independent samples
paired samples
same unit, observed twice
E.g., ask Bob, Jane, Nathan and Claire how they feel about xyz at
time t. Then ask them again at time t + 1. Same people, but at a
different time
independent samples
Independent samples: different units (e.g., different individuals)
There is no matching of the units in the two samples, and the two
samples may be of different sizes.
The null hypothesis is that there is NO difference between the two
samples in the variable of interest (e.g. grades).
