Provide Information Regarding Statistics & Econometrics : Introduction to t Distribution Lecture 41

Introduction

t-Statistic

Lecture 41

Introduction

The z statistic is used as a test statistic to make inferences about the population mean or the difference between population means (to construct a confidence interval for the population mean, the difference of the population means, or to test the hypothesis about the population means, the difference between the population means) when the population standard deviation is known or unknown and the sample size is large. Now, if making inferences about the population mean or the difference between population means and the population standard deviation(s) is unknown and the sample size is small, the t statistic is used as a test statistic. The t statistic is also known as the student’s t statistic.

The student’s t statistic was developed by William Sealy Gosset in 1908 using the nickname Student. William Sealy also developed the t-test and t-distribution. William Sealy, who worked at the Guinness brewery in Dublin, discovered that the small sample sizes he experienced at work were inappropriate for the statistical methods that were then in use, which used large samples. The student develops the following test statistic to tackle the small sample(s).

Where:

ν: Sample size minus population parameters to be estimated.

n: Sample size.

s: Unbiased estimate of population standard deviation.

Assumptions using Student's t distribution

i. A random selection process will be employed when selecting the sample observations.

ii. The sampled population should be normal. However, a slight departure from normality does not seriously affect the test.

iii. In the case of two samples, both samples are selected randomly and independently from two normal populations having identical variances.

Sampling Distribution of t

The t variable is the ratio of the standard normal variable and the square root of the chi square variable divided by its degree of freedom.

Let X1, X2,..., Xn be the observations of a random sample of size n drawn from a normal distribution with mean μ and standard deviation σ.

The sampling distribution of t is called t-distribution with (n-1) degree of freedom. The probability function of t is given by

Properties of t-distribution

i. The t-distribution is symmetric and continuous around t = 0 ranges from $- \propto t o + \propto.$

ii. The mean of t-distribution is equal to zero.

μt =E(t)=0.

iii. The variance of the t-distribution is dependent on the degree of freedom and equal to v/v-2.

σt =Var(t)= v / v-2

iv. The t-distribution is uni-model, and mode of t-distribution is equal to zero.

v. The shape of the t-distribution curve is identical to the normal distribution curve but flatter than the normal curve for a small sample size.

vi. The shape of the t-distribution approach to the normal curve if the sample size is sufficiently large.

vii. The student's t distribution is independent of parameters and depends on the degree of freedom.

vii. The student's t distribution is symmetrical about zero. thus t1-α =- tα

The shape of the student's t distribution and normal distribution curves are given below:

Small Samples Confidence Interval

Confidence Interval for the Mean of a Normal Population When n is small and σ is unknown.

Let X¯ be the unbiased estimate of μ computed from the values of a small random sample of size n selected from a normal population having mean "μ" and unknown standard deviation "σ". As the sample size is small, the sampling distribution of X¯ approaches the t distribution with v = n -1 degree of freedom.

Thats