Question

Estimating the Expectation A measurement follows the normal distribution with a standard deviation of 15 and an unknown expectation u. You can consider that measurement to be the "original distribution. Two statisticians propose two distinct ways to estimate the unknown quantity u with the aid of a sample of size 36. They will do that by evaluating two different SAMPLING DISTRIBUTIONS to determine which method is better There are two statisticians involved in this task: statistician "A" and statistician "B Statistician A proposes to use the sampling distribution of the sample average as an estimate. Statistician B proposes to use the sampling distribution of the sample median instead In order to choose between the two options they agree to prefer the statistic that has a smaller variance (with respect to the sampling distribution)

How do I get the unknown expectation for the two different sampling method chosen by the two statisticians?

Answer 1

Let the measurement be X

X ~ Normal distribution with mean $\large \mu$ and standard deviation = $\large \sigma$ = 15

Consider

A. Sampling distribution of the sample average

     The sampling distribution of the sample average of X is a Normal Distribution

     with mean $\large \mu$ and standard deviation =

σ1Vn = 151 36 = 2.5

     Thus variance for sampling distribution of the sample average = (2.5)² = 6.25

B. Sampling distribution of the sample median

     The sampling distribution of the sample median of X is a Normal Distribution

     with mean $\large \mu$ and standard deviation =

15 1.253-_-1.253 Tl - 36-3.125

     Thus variance for sampling distribution of the sample median = (3.1325)² = 9.8126

We can see from A and B, that the variance for sampling distribution of the sample average is smaller among

the two variances.

Thus Statistician A's sampling distribution of sample average is the preferred estimate