Probstat/week15 practice 1

This is part of probstat.

Data generation

Generate 5 sets of samples from normal populations, each of size 10, on a spreadsheet software. Make sure that the mean and variance of each set can be modified easily.

Variance estimations

Estimator 1. The first estimator for $\sigma ^{2}$ is

${\frac {1}{m}}\sum _{i=1}^{m}\sum _{j=1}^{n}{\frac {(X_{ij}-{\bar {X}}_{i\centerdot })^{2}}{n-1}}.$

This estimator is always correct. (Even when $H_{0}$ is not true.)

Estimator 2. The second estimator is calculated from the sample means. It will be correct only when $H_{0}$ is true.

$n\sum _{i=1}^{m}{\frac {({\bar {X}}_{i\centerdot }-{\bar {X}}_{\centerdot \centerdot })^{2}}{m-1}}$

Hypothesis testing

Then compute

$SS_{w}=\sum _{i=1}^{m}\sum _{j=1}^{n}(X_{ij}-{\bar {X}}_{i\centerdot })^{2}$

and

$SS_{b}=n\sum _{i=1}^{m}({\bar {X}}_{i\centerdot }-{\bar {X}}_{\centerdot \centerdot })^{2}$

Recall that

${\frac {SS_{w}}{\sigma ^{2}}}\sim \chi _{m(n-1)}^{2}$,

and

${\frac {SS_{b}}{\sigma ^{2}}}\sim \chi _{m-1}^{2}$.

Also, since

Error

Too many requests (f061ab2)

If you report this error to the Wikimedia System Administrators, please include the details below.

Request served via cp5021 cp5021, Varnish XID 686639132
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Fri, 13 Mar 2026 10:01:01 GMT


Sensitive client information

IP address: 158.108.32.49

$S_{i}^{2}$ is independent of

Error

Too many requests (f061ab2)

If you report this error to the Wikimedia System Administrators, please include the details below.

Request served via cp5021 cp5021, Varnish XID 724210526
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Fri, 13 Mar 2026 10:01:01 GMT


Sensitive client information

IP address: 158.108.32.49

${\bar {X}}_{i\centerdot }$,

Error

Too many requests (f061ab2)

If you report this error to the Wikimedia System Administrators, please include the details below.

Request served via cp5021 cp5021, Varnish XID 709437103
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Fri, 13 Mar 2026 10:01:01 GMT


Sensitive client information

IP address: 158.108.32.49

$SS_{w}$ and $SS_{b}$ are independent. Thus, the statistic

Error

Too many requests (f061ab2)

If you report this error to the Wikimedia System Administrators, please include the details below.

Request served via cp5021 cp5021, Varnish XID 682478875
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Fri, 13 Mar 2026 10:01:02 GMT


Sensitive client information

IP address: 158.108.32.49

${\frac {SS_{b}/(m-1)}{SS_{w}/m(n-1)}}\sim F_{m-1,m(n-1)}.$

Therefore, we can use the table for the F-distribution to test for $H_{0}$.