[FONT="]1. This question has multiple parts. Assume X is a normally distributed random [/FONT]

[FONT="]variable with mean µ and standard deviation σ. A sample of size n = 5 from this [/FONT]

[FONT="]distribution is given as (−1.5, 1, 0.5, 3.2, 4.8). State two estimators for each of the [/FONT]

[FONT="]listed parameters below. Choose one of the estimators, say ˆθ to be the standard [/FONT]

[FONT="]estimator that we have seen in class (for example the sample mean is a good [/FONT]

[FONT="]estimator for the parameter population mean). Let the second estimator be ˜θ = [/FONT]

[FONT="]ˆθ + 1. Use both estimators ˆθ and ˜θ, and the sample given above to obtain two [/FONT]

[FONT="]estimates, one with each estimator for each of the following parameters. Establish [/FONT]

[FONT="]the Bias, Variance, and the Mean Squared Error (MSE) analytically.

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[FONT="]a. The mean of the sampling distribution of the sample sum [/FONT]

[FONT="]b. The standard deviation (or standard error) of the sampling distribution of the [/FONT]

[FONT="]sample mean [/FONT]

[FONT="]c. The standard deviation (or standard error) of the sampling distribution of the [/FONT]

[FONT="]sample sum[/FONT]