Introduction to statistics: The sampling distribution, ttest · The sampling distribution of the...
Embed Size (px)
Transcript of Introduction to statistics: The sampling distribution, ttest · The sampling distribution of the...

1/ 36
Lecture 2
Introduction to statistics: The samplingdistribution, ttest
Shravan Vasishth
Universität [email protected]
http://www.ling.unipotsdam.de/∼vasishth
April 10, 2020
1 / 36

2/ 36
Lecture 2
The sampling distribution of the mean
Sampling from the normal distribution
The sampling distribution of the mean
When we have a single sample, we know how to compute MLEsof the sample mean and standard deviation, µ̂ and σ̂.Suppose now that you had many repeated samples; from eachsample, you can compute the mean each time. We can simulatethis situation:
x

3/ 36
Lecture 2
The sampling distribution of the mean
Sampling from the normal distribution
The sampling distribution of the mean
Let’s repeatedly simulate sampling 1000 times:
nsim

4/ 36
Lecture 2
The sampling distribution of the mean
Sampling from the normal distribution
The sampling distribution of the mean
Plot the distribution of the means under repeated sampling:
Samp. distrn. means
µ̂
density
470 480 490 500 510 520 530
0.0
00.0
10.0
20.0
30.0
4
4 / 36

5/ 36
Lecture 2
The sampling distribution of the mean
Sampling from the exponential distribution
The sampling distribution of the mean
Interestingly, it is not necessary that the distribution that we aresampling from be the normal distribution.
for(i in 1:nsim){x

6/ 36
Lecture 2
The sampling distribution of the mean
Sampling from the exponential distribution
The sampling distribution of the mean
Samp. distrn. means
µ̂
density
0.8 1.0 1.2 1.4
01
23
6 / 36

7/ 36
Lecture 2
The sampling distribution of the mean
The central limit theorem
The central limit theorem
1. For large enough sample sizes, the sampling distribution of themeans will be approximately normal, regardless of theunderlying distribution (as long as this distribution has a meanand variance defined for it).
2. This will be the basis for statistical inference.
7 / 36

8/ 36
Lecture 2
The sampling distribution of the mean
Standard error
The sampling distribution of the mean
We can compute the standard deviation of the samplingdistribution of means:
## estimate from simulation:
sd(samp_distrn_means)
## [1] 0.10191
8 / 36

9/ 36
Lecture 2
The sampling distribution of the mean
Standard error
The sampling distribution of the mean
A further interesting fact is that we can compute this standarddeviation of the sampling distribution from a single sample ofsize n:σ̂√n
x

10/ 36
Lecture 2
The sampling distribution of the mean
Standard error
The sampling distribution of the mean
1. So, from a sample of size n, and sd σ or an MLE σ̂, we cancomputethe standard deviation of the sampling distribution of themeans.
2. We will call this standard deviation the estimated standarderror.SE = σ̂√
n
I say estimated because we are estimating SE using an anestimate of σ.
10 / 36

11/ 36
Lecture 2
The sampling distribution of the mean
Confidence intervals
Confidence intervals
The standard error allows us to define a socalled 95% confidenceinterval:
µ̂± 2SE (1)
So, for the mean, we define a 95% confidence interval as follows:
µ̂± 2 σ̂√n
(2)
11 / 36

12/ 36
Lecture 2
The sampling distribution of the mean
Confidence intervals
Confidence intervals
In our example:
## lower bound:
mu(2*hat_sigma/sqrt(n))
## [1] 480.03
## upper bound:
mu+(2*hat_sigma/sqrt(n))
## [1] 519.97
12 / 36

13/ 36
Lecture 2
The sampling distribution of the mean
Confidence intervals
The meaning of the 95% CI
If you take repeated samples and compute the CI each time, 95%of those CIs will contain the true population mean.
lower

14/ 36
Lecture 2
The sampling distribution of the mean
Confidence intervals
The meaning of the 95% CI
## check how many CIs contain mu:
CIs

15/ 36
Lecture 2
The sampling distribution of the mean
Confidence intervals
The meaning of the 95% CI
0 20 40 60 80 100
56
58
60
62
64
95% CIs in 100 repeated samples
i−th repeated sample
Score
s
15 / 36

16/ 36
Lecture 2
The sampling distribution of the mean
Confidence intervals
The meaning of the 95% CI
1. The 95% CI from a particular sample does not mean that theprobability that the true value of the mean lies inside thatparticular CI.
2. Thus, the CI has a very confusing and (not very useful!)interpretation.
3. In Bayesian statistics we use the credible interval, which has amuch more sensible interpretation.
However, for large sample sizes, the credible and confidenceintervals tend to be essentially identical.For this reason, the CI is often treated (this is technicallyincorrect!) as a way to characterize uncertainty about our estimateof the mean.
16 / 36

17/ 36
Lecture 2
The sampling distribution of the mean
Confidence intervals
Main points from this lecture
1. We compute maximum likelihood estimates of the meanx̄ = µ̂ and standard deviation σ̂ to get estimates of the truebut unknown parameters.
x̄ =∑n
i=1 xin
2. For a given sample, having estimated σ̂, we estimate thestandard error:SE = σ̂/
√n
3. This allows us to define a 95% CI about the estimated mean:µ̂± 2× SE
From here, we move on to statistical inference and null hypothesissignificance testing (NHST).
17 / 36

18/ 36
Lecture 2
The story so far
1. We defined random variables.
2. We learnt about pdfs and cdfs, and learnt how to computeP (X < x).
3. We learnt about Maximum Likelihood Estimation.
4. We learnt about the sampling distribution of the samplemeans.
This prepares the way for null hypothesis significance testing(NHST).
18 / 36

19/ 36
Lecture 2
Statistical inference
Hypothesis testing
Suppose we have a random sample of size n, and the data comefrom a N(µ, σ) distribution.We can estimate sample mean x̄ = µ̂ and σ̂, which in turn allowsus to estimate the sampling distribution of the mean under(hypothetical) repeated sampling:
N(x̄,σ̂√n
) (3)
19 / 36

20/ 36
Lecture 2
Statistical inference
The onesample hypothesis test
Imagine taking an independent random sample from a randomvariable X that is normally distributed, with mean 12 and standarddeviation 10, sample size 11. We estimate the mean and SE:
sample

21/ 36
Lecture 2
Statistical inference
The onesample ttest
The onesample test
The NHST approach is to set up a null hypothesis that µ has somefixed value. For example:
H0 : µ = 0 (4)
This amounts to assuming that the true distribution of samplemeans is (approximately*) normally distributed and centeredaround 0, with the standard error estimated from the data.
* I will make this more precise in a minute.
21 / 36

22/ 36
Lecture 2
Statistical inference
The onesample ttest
Null hypothesis distribution
−20 −10 0 10 20
0.0
0.1
0.2
0.3
0.4
x
density
sample mean
22 / 36

23/ 36
Lecture 2
Statistical inference
The onesample ttest
NHST
The intuitive idea is that
1. if the sample mean x̄ is near the hypothesized µ (here, 0), thedata are (possibly) “consistent with” the null hypothesisdistribution.
2. if the sample mean x̄ is far from the hypothesized µ, the dataare inconsistent with the null hypothesis distribution.
We formalize “near” and “far” by determining how many standarderrors the sample mean is from the hypothesized mean:
t× SE = x̄− µ (5)
This quantifies the distance of sample mean from µ in SE units.
23 / 36

24/ 36
Lecture 2
Statistical inference
The onesample ttest
NHST
So, given a sample and null hypothesis mean µ, we can computethe quantity:
t =x̄− µSE
(6)
Call this the tvalue. Its relevance will just become clear.
24 / 36

25/ 36
Lecture 2
Statistical inference
The onesample ttest
NHST
The quantity
T =X̄ − µSE
(7)
has a tdistribution, which is defined in terms of the sample size n.We will express this as: T ∼ t(n− 1)Note also that, for large n, T ∼ N(0, 1).
25 / 36

26/ 36
Lecture 2
Statistical inference
The onesample ttest
NHST
Thus, given a sample size n, and given our null hypothesis, we candraw tdistribution corresponding to the null hypothesisdistribution.For large n, we could even use N(0,1), although it is traditional inpsychology and linguistics to always use the tdistribution nomatter how large n is.
26 / 36

27/ 36
Lecture 2
Statistical inference
The onesample ttest
The tdistribution vs the normal
1. The tdistribution takes as parameter the degrees of freedomn− 1, where n is the sample size (cf. the normal, which takesthe mean and variance/standard deviation).
2. The tdistribution has fatter tails than the normal for small n,say n < 20, but for large n, the tdistribution and the normalare essentially identical.
27 / 36

28/ 36
Lecture 2
Statistical inference
The onesample ttest
The tdistribution vs the normal
−4 −2 0 2 4
0.0
0.1
0.2
0.3
0.4
df= 2
−4 −2 0 2 4
0.0
0.1
0.2
0.3
0.4
df= 5
−4 −2 0 2 4
0.0
0.1
0.2
0.3
0.4
df= 15
−4 −2 0 2 4
0.0
0.1
0.2
0.3
0.4
df= 20
28 / 36

29/ 36
Lecture 2
Statistical inference
The onesample ttest
ttest: Rejection region
So, the null hypothesis testing procedure is:
1. Define the null hypothesis: for example, H0 : µ = 0.
2. Given data of size n, estimate x̄, standard deviation s,standard error s/
√n.
3. Compute the tvalue:
t =x̄− µs/√n
(8)
4. Reject null hypothesis if tvalue is large (to be made moreprecise next).
29 / 36

30/ 36
Lecture 2
Statistical inference
The onesample ttest
ttest
How to decide when to reject the null hypothesis? Intuitively, whenthe tvalue from the sample is so large that we end up far in eithertail of the distribution.
30 / 36

31/ 36
Lecture 2
Statistical inference
The onesample ttest
ttest
−20 −10 0 10 20
0.0
0.1
0.2
0.3
0.4
t(n−1)
x
density
sample mean
31 / 36

32/ 36
Lecture 2
Statistical inference
The onesample ttest
Rejection region
1. For a given sample size n, we can identify the “rejectionregion” by using the qt function (see lecture 1).
2. Because the shape of the tdistribution depends on the degreesof freedom (n1), the critical tvalue beyond which we rejectthe null hypothesis will change depending on sample size.
3. For large sample sizes, say n > 50, the rejection point isabout 2.
abs(qt(0.025,df=15))
## [1] 2.1314
abs(qt(0.025,df=50))
## [1] 2.0086
32 / 36

33/ 36
Lecture 2
Statistical inference
The onesample ttest
ttest: Rejection region
Consider the tvalue from our sample in our running example:
## null hypothesis mean:
mu

34/ 36
Lecture 2
Statistical inference
The onesample ttest
ttest: Rejection region
So, for large sample sizes, if  t > 2 (approximately), we can rejectthe null hypothesis.For a smaller sample size n, you can compute the exact criticaltvalue:
qt(0.025,df=n1)
This is the critical tvalue on the lefthand side of thetdistribution. The corresponding value on the righthand side is:
qt(0.975,df=n1)
Their absolute values are of course identical (the distribution issymmetric).
34 / 36

35/ 36
Lecture 2
Statistical inference
The onesample ttest
The tdistribution vs the normal
Given the relevant degrees of freedom, one can compute the areaunder the curve as for the Normal distribution:
pt(2,df=10)
## [1] 0.036694
pt(2,df=20)
## [1] 0.029633
pt(2,df=50)
## [1] 0.025474
Notice that with large degrees of freedom, the area under thecurve to the left of 2 is approximately 0.025.
35 / 36

36/ 36
Lecture 2
Statistical inference
The onesample ttest
The t.test function
The t.test function in R delivers the tvalue:
## from ttest function:
## tvalue
t.test(sample)$statistic
## t
## 1.781
36 / 36
The sampling distribution of the meanSampling from the normal distributionSampling from the exponential distributionThe central limit theoremStandard errorConfidence intervals
The story so farStatistical inferenceThe onesample ttest