Statistics

interval estimation of population mean with  σ is known;
Step 1: Check requirements
Sample is simple random sample
Either sample size n>30 or sample is normally distributed
Standard deviation for the population  is σ known
Step 2: Estimate interval

We use  as the critical value, because we assume that  the sample mean is normally distributed around the population mean

Estimating a Population Mean:σ not known

If we do not know the value of the population’s Standard deviation 
We need to use Student’s t-distribution instead of the normal distribution that we used as an assumption for interval estimation

Since, we do not know the standard deviation of the population we estimate it with the value of the sample standard deviation s, but this introduces another source of unreliability. In order to maintain a desired confidence level, we compensate for this additional unreliability by making the confidence interval wider.

Developed by William Gosset at the Guinness Brewery in Dublin, who wrote under the pseudonym ‘Student’ 
If a population has a normal distribution, the distribution of all samples of size n collected from the population forms:

It is a Student’s t-Distribution for all samples size n which is often referred simply as a t-distribution. 

Important Properties of the t-Distribution
1. It is different for different sample sizes
2. Same general bell shape as the standard normal distribution, but it reflects the greater variability that is expected with small samples
3.It has a mean of t = 0 
4.The standard deviation of the Student t-distribution varies with the sample size, but it is greater than 1
5.As the sample size n gets larger, the Student t-distribution gets closer to the standard normal
distribution

Interval estimation of population mean with  is unknown
Step 1: Check requirements
- Sample is simple random sample
- Either sample size n>30 or sample is normally distributed 
- Standard deviation for the population σ is unknown
Step 2: Confidence interval

We use 𝑡_(𝛼/2) as the critical value, because we assume that the sample mean is consistent with  a Student t-distribution around the population mean .
 

In order to facilitate the interval estimation, mathematicians and statistics pioneers developed a t-table which lists the corresponding critical values against the significance level and the degree of freedom of sample used for estimation.
Confidence level: 1 − 𝛼
Significance level: 𝛼
Degree of freedom: n − 1 ( is sample size)
t-tables are used to provide a t-score 
Degrees of Freedom is given in the left-hand column, and the body of the table tests the level of significance for either one-tailed test or two-tailed test. The table is shown below.

Degrees of Freedom

The number of degrees of freedom for a collection of sample data is the number of sample values that can vary after certain restrictions have been imposed on all data values.
Often abbreviated as df
where df = n – 1
Example: If 10 students have quiz scores with a mean of 80, we can freely assign values to the first 9 scores, but the 10 the score is then determined by the remainder

10th. score = 800 – (score 1 + score 2 + score 3 +…+ score 9) Because the first 9 scores may be freely selected to be any values, we say there are 9 degrees of freedom available.

Finding a Critical t-Value

Example:
A sample of size n = 7 is a simple random sample selected from a normally distributed population

Find the critical value  𝑡_(𝛼/2)  corresponding to a 95% confidence level
Because n = 7, the number of degrees of freedom = n – 1 = 6
A 95% confidence level corresponds to α = 0.05, and as we are looking for a confidence interval this
considers both tails, so we find the column listing values for an area of 0.05 in two tails

Answer:

Degrees of Freedom = 6

α = 0.05

t_(α/2) = 2.447

This can also be expressed as:

t_0.025 = 2.447

Procedure:
1. Verify that the requirements are satisfied
2. Using n – 1 degrees of freedom, refer to the t tables and find the critical value  𝑡_(𝛼/2), that corresponds to the desired confidence level
3. Evaluate the margin of error 

4. Using the sample mean and margin of error, find values for the confidence interval