Population Mean: Small Sample Case



10.2. Population Mean: Small Sample Case

t-distribution (“student” t distribution):

This distribution was invented by W. S. Gossett (published in the name “student”). The t-distribution is a family of similar probability distributions, with a specific t distribution depending on a parameter known the degree of freedom (d.f.). Denote [pic] to be the random variable having t-distribution with n degree of freedom.

Example:

[pic]; [pic]

[pic] [pic]

[pic]

Note: [pic]. The t-distribution is symmetric about 0.

[pic]: a t value with an area of [pic] in the upper tail of the t-distribution with degree of freedom equal to n.

[pic].

Example:

[pic]

[pic]

Example:

For a t distribution with 16 degrees of freedom, find the area of probability.

(a) To the left of -1.746.

(b) Between -1.337 and 2.120.

[solution:]

(a)

[pic].

(b)

[pic]

Important Result:

When the population has a normal probability distribution and [pic] is

unknown, then

[pic],

where [pic], [pic], and [pic] are random

variables with associated possible values [pic].

Derivation of [pic] confidence interval:

Suppose the population has a normal probability distribution. Since

[pic]

thus

[pic]

[pic] confidence interval based on t-distribution:

As the population has a normal distribution and [pic] is unknown,

[pic]

is a [pic] confidence interval of [pic].

Example:

Consider the following random sample of 4 observations, 25, 47, 32, 56. Suppose the population is normally distributed. Please construct a 95% confidence interval for [pic].

[solution:]

[pic]

In addition, [pic] and [pic]. Thus,

[pic]

Example:

Suppose we have the following data from a normal population

|50 |48 |55 |52 |53 |46 |54 |50 |

Provide a 95% confidence interval for the population mean.

[solution:]

[pic] Then, a 95% confidence interval is

[pic]Online Exercise:

Exercise 10.2.1

Exercise 10.2.2

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