Question

HW6#14

Three randomly selected households are surveyed. The numbers of people in the households are 2, 3, and 10.

Assume that samples of size n=2

are randomly selected with replacement from the population of 2, 3,and 10.

Listed below are the nine different samples. Complete parts (a) through (c).

2,2

2,3

2,10

3,2

3,3

3,10

10,2

10,3

10,10

A. Find the median of each of the nine samples, then summarize the sampling distribution of the medians in the format of a table representing the probability distribution.

B. Compare the population median to the mean of the sample medians.

C. Do the sample medians target the value of the population median? In general, do sample medians make good estimators of population medians? why or why not?

Answer #1

A.

Samples | Sample median |

2,2 | 2 |

2,3 | 2.5 |

2,10 | 6 |

3,2 | 2.5 |

3,3 | 3 |

3,10 | 6.5 |

10,2 | 6 |

10,3 | 6.5 |

10,10 | 10 |

The sampling distribution of the medians is as follows:

Sample medians | Probability |

2 | 1/9 |

2.5 | 2/9 |

3 | 1/9 |

6 | 2/9 |

6.5 | 2/9 |

10 | 1/9 |

B. Mean of the sample medians=2*(1/9)+2.5*(2/9)+3*(1/9)+6*(2/9)+6.5*(2/9)+10*(1/9)=5

Population median=3

Mean of sample medians>Population median.

C. Since sample median is not an unbiased estimator of population median so the sample medians do not target the value of the population median.

In general, sample medians do not make good estimators of population medians since it is a biased estimator of population median. However sample mean is good estimator of population mean since it is an unbiased estimator of population mean.

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