# Statistics problem that revolves around Estimators and Sampling distributions

#### carl0545

##### New member
[FONT=&quot]I do not currently have a strong grasp on these statistical concept yet. If someone could walk me through this problem it will help me considerably with the rest of my homework assignment. [/FONT]

[FONT=&quot]1. This question has multiple parts. Assume X is a normally distributed random [/FONT]
[FONT=&quot]variable with mean µ and standard deviation σ. A sample of size n = 5 from this [/FONT]
[FONT=&quot]distribution is given as (−1.5, 1, 0.5, 3.2, 4.8). State two estimators for each of the [/FONT]
[FONT=&quot]listed parameters below. Choose one of the estimators, say ˆθ to be the standard [/FONT]
[FONT=&quot]estimator that we have seen in class (for example the sample mean is a good [/FONT]
[FONT=&quot]estimator for the parameter population mean). Let the second estimator be ˜θ = [/FONT]
[FONT=&quot]ˆθ + 1. Use both estimators ˆθ and ˜θ, and the sample given above to obtain two [/FONT]
[FONT=&quot]estimates, one with each estimator for each of the following parameters. Establish [/FONT]
[FONT=&quot]the Bias, Variance, and the Mean Squared Error (MSE) analytically.
[/FONT]

[FONT=&quot]a. The mean of the sampling distribution of the sample sum [/FONT]
[FONT=&quot]b. The standard deviation (or standard error) of the sampling distribution of the [/FONT]
[FONT=&quot]sample mean [/FONT]
[FONT=&quot]c. The standard deviation (or standard error) of the sampling distribution of the [/FONT]
[FONT=&quot]sample sum[/FONT]

#### Subhotosh Khan

##### Super Moderator
Staff member
I do not currently have a strong grasp on these statistical concept yet. If someone could walk me through this problem it will help me considerably with the rest of my homework assignment.

1. This question has multiple parts. Assume X is a normally distributed random
variable
with mean µ and standard deviation σ. A sample of size n = 5 from this
distribution is given as (−1.5, 1, 0.5, 3.2, 4.8). State two estimators for each of the
listed parameters below. Choose one of the estimators, say ˆθ to be the standard
estimator that we have seen in class (for example the sample mean is a good
estimator for the parameter population mean). Let the second estimator be ˜θ =
ˆθ + 1. Use both estimators ˆθ and ˜θ, and the sample given above to obtain two
estimates, one with each estimator for each of the following parameters. Establish
the Bias, Variance, and the Mean Squared Error (MSE) analytically.

a. The mean of the sampling distribution of the sample sum
b. The standard deviation (or standard error) of the sampling distribution of the
sample mean
c. The standard deviation (or standard error) of the sampling distribution of the
sample sum

• the definitions of the parameters marked in red above.
• difference between sample mean and population mean.