Question

The following sample of n = 6 scores was obtained from a population with unknown parameters.

Scores: 7, 1, 6, 3, 6, 7

a. Compute the sample mean and standard deviation.

(Note that these are descriptive values that summarize the sample data.)

b. Compute the estimated standard error for M.

(Note that this is an inferential value that describes how accurately the sample mean represents the unknown population mean.)

Answer 1

a)
`\bar x =(7+1+6+3+6+7)/(6)=5` represent the mean

`s=\sqrt{((7-5)^2 + (1-5)^2 +(6-5)^2 +(3-5)^2 +(6-5)^2 + (7-5)^2)/(6-1)}=2.449` represent the sample standard deviation

b)
`SE= (2.449)/(√(6))= 0.9998`

Explanation:

Previous concepts

The margin of error is the range of values below and above the sample statistic in a confidence interval.

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

The central limit theorem states that "if we have a population with mean μ and standard deviation σ and take sufficiently large random samples from the population with replacement, then the distribution of the sample means will be approximately normally distributed. This will hold true regardless of whether the source population is normal or skewed, provided the sample size is sufficiently large".

Solution to the problem

Assuming the X follows a normal distribution

`X \sim N(\mu, \sigma)`

From the central limit theorem the distribution for the sample mean is given by:

`\bar X \sim N(\mu, (\sigma)/(√(n)))`

Data given: 7, 1, 6, 3, 6, 7

Part a

We can calculate the sample mean and deviation with the following formulas:

`\bar X = (\sum_(i=1)^n X_i)/(n)`

`s=\sqrt{(\sum_(i=1)^n (X_i -\bar X)^2)/(n-1)}`

`\bar x =(7+1+6+3+6+7)/(6)=5` represent the mean

`s=\sqrt{((7-5)^2 + (1-5)^2 +(6-5)^2 +(3-5)^2 +(6-5)^2 + (7-5)^2)/(6-1)}=2.449` represent the sample standard deviation

Part b

From the central limit theorem we know that the standard error is given by:

`SE= (\sigma)/(√(n)))`

And the estimation for the standard error is given by:

`\hat{SE}= (s)/(√(n)))`

And replacing the values we got:

`SE= (2.449)/(√(6))= 0.9998`