Refer to Exercise 5.3.

a. Find the sampling distribution of s2.

b. Find the population variance σ2.

c. Show that s2 is an unbiased estimator of σ2.

d. Find the sampling distribution of the sample standard deviation s.

e. Show that s is a biased estimator of σ.

Step 1 of 5</p>

a) We have to find the sampling distribution of s2

From the data given in exercise we have find s2

Now calculate the s2 values for each sample

For the sample (1,1)

Then s2 =[(1-1)2+(1-1)2]/1

=0

For the sample (1,2)

Then s2 =[(1-15)2+(2-1.5)2]/1

=0.5

Calculate the s2 values for all the remaining samples

s2 |
p(s2) |
s2 |
p(s2) |

0 |
0.04 |
0.5 |
0.04 |

0.5 |
0.06 |
2 |
0.02 |

2 |
0.04 |
4.5 |
0.04 |

4.5 |
0.04 |
2 |
0.06 |

8 |
0.02 |
0.5 |
0.04 |

0.5 |
0.06 |
0 |
0.04 |

0 |
0.09 |
0.5 |
0.02 |

0.5 |
0.06 |
8 |
0.02 |

2 |
0.06 |
4.5 |
0.03 |

4.5 |
0.03 |
2 |
0.02 |

2 |
0.04 |
0.5 |
0.02 |

0.5 |
0.06 |
0 |
0.01 |

0 |
0.04 |

Now the sampling distribution of s2 is

s2 |
0 |
0.5 |
2 |
4.5 |
8 |

p(s2) |
0.22 |
0.36 |
0.24 |
0.14 |
0.04 |

Step 2 of 5</p>

b) We have to find population variance

Now

=

=1.61