# 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
Please tell us

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

#### carl0545

##### New member
Please tell us

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

Can do, and thank you for the response. My understanding of a normally distributed variable is that it is based on the variable's mean and standard deviation. Looking at a graph of a normally distributed variable showcases a bell curve, with the peak of the curve representing the mean along the x-axis. The standard deviation of a set of numbers of is the average amount of variance between them. For instance the set (1,3,2,4,5) would have a much smaller standard deviation than (1,321,-760,340,50). Now we have just started to learn about estimators, but my understanding of them is that they are a statistic (so created from a sample not the population) to give an estimate about the population. So the sample mean can be an estimator to find an estimate of the population mean. I believe. The Bias, Variance, and MSE are essentially assessing the accuracy and precision of the estimator as compared to the true parameter on average. The Bias of an estimator shows us how far off the estimator is from the true value. The Variance of an estimator, I am a little confused by but I believe it show how much the estimates vary from one another. A smaller variance is preferred to a larger one. MSE is a combination of the Bias and Variance.

The population mean is the average taken from an entire population of numbers, while the sample mean is the average taken from a sample from the population.

Thank you again for helping me out!