Two Types of Estimation

One of the major applications of statistics is estimating population parameters from sample statistics. There are types of estimation:

• Point Estimate: the value of sample statistics

Point estimates of average height with multiple samples (Source: Zhihu)

• Confidence Intervals: intervals constructed using a method that contains the population parameter a specified proportion of the time.

95% confidence interval of average height with multiple samples (Source: Zhihu)

 

Confidence Interval for the Mean

Population Variance is known

Suppose that M is the mean of N samples X₁, X₂, ......, X_n, i.e.

 

According to Central Limit Theorem, the the sampling distribution of the mean M is 

where μ and σ² are the mean and variance of the population respectively. If repeated samples were taken and the 95% confidence interval computed for each sample, 95% of the intervals would contain the population mean. So the 95% confidence interval for M is the inverval that is symetric about the point estimate μ so that the area under normal distribution is 0.95.

That is,

Since we don't know the mean of population, we could use the sample mean  instead.

Population Variance is Unknown

Dregree of Freedom

The degrees of freedom (df) of an estimate is the number of independent pieces of information on which the estimate is based. In general, the degrees of freedom for an estimate is equal to the number of values minus the number of parameters estimated en route to the estimate in question. 

If the variance in a sample is used to estimate the variance in a population, we couldn't calculate the sample variace as

That's because we have two parameters to estimate (i.e., sample mean and sample variance). The degree of freedom should be N-1, so the previous formula underestimates the variance. Instead, we should use the following formula 

where s² is the estimate of the variance and M is the sample mean. The denominator of this formula is the degree of freedom. 

Student's t-Distribution 

Suppose that X is a random variable of normal distribution, i.e., X ~ N(μ, σ²)

is sample mean and

is sample deviation.

 

is a random variable of normal distribution.

is a random variable of student's t distribution.

The probability density function of T is

where  is the degree of freedom,  is a gamma function.