1 More about the Sampling Distribution of the Sample Mean and introduction to the t-distribution Presentation 3

2 Brief Review of Sampling Distributions If the sample is large enough, np and n(1-p) >5, then If the sample is large enough, np and n(1-p) >5, then Sampling Distribution of Sample Proportion: Each member of the population has a trait of interest with probability p (population proportion). Each member of the population has a trait of interest with probability p (population proportion). Suppose a random sample of size n is obtained form the population. Suppose a random sample of size n is obtained form the population. The sample proportion p-hat is a logical estimator of p, The sample proportion p-hat is a logical estimator of p,

3 Sampling Distribution of Sample Mean Let X be a random variable and the statistic be the sample mean of X in a random sample of size n. Let X be a random variable and the statistic be the sample mean of X in a random sample of size n. We examine the sampling distribution of in the following three scenarios: We examine the sampling distribution of in the following three scenarios: 1. When X is a normal random variable, E(X)=µ and s.d.(X)= σ are both known. 2. When X is not a normal random variable, E(X)=µ and s.d.(X)= σ are both known and the sample size is large, n ≥ When X is a normal random variable, E(X)=µ is known, s.d.(X)= σ is unknown and the sample size is large, n ≥ 30. So far we have seen the first two cases. So far we have seen the first two cases.

4 Case 1: Case 1: Case 2: Case 2: Case 3: Case 3:, where t n-1 denotes the t-distribution with n -1 degrees of freedom., where t n-1 denotes the t-distribution with n -1 degrees of freedom.

5 Properties of the t-distribution There are infinitely many t-distributions, each characterized by one parameter, the degrees of freedom. There are infinitely many t-distributions, each characterized by one parameter, the degrees of freedom. The degrees of freedom are positive integers, e.g. t 1, t 2, t 3,…, t 10,… The degrees of freedom are positive integers, e.g. t 1, t 2, t 3,…, t 10,… Random variables with t-distribution are continuous. Random variables with t-distribution are continuous. The density curve of a t - distribution is symmetric, bell-shaped and centered at zero (similar to the standard normal curve). The density curve of a t - distribution is symmetric, bell-shaped and centered at zero (similar to the standard normal curve). There are tables for the probabilities related with a t – random variable. In will see how to use them later in the course. There are tables for the probabilities related with a t – random variable. In will see how to use them later in the course. As the degrees of freedom increase, the variance of the t -random variable decreases, i.e. the density curve is less spread, and actually it approaches the standard normal density. As the degrees of freedom increase, the variance of the t -random variable decreases, i.e. the density curve is less spread, and actually it approaches the standard normal density.

6 Properties of the t-distribution Z~N(0,1) t 10 t3t3 t1t1