# Lesson 5: Sample Mean Vector and Sample Correlation and Related Inference Problems

### Introduction

In this lesson we consider the properties of the sample mean vector and the sample correlations which we had defined earlier. We will also consider estimation and hypothesis testing problems on population mean and correlation coefficients.

### Lesson Objectives:

Upon completion of this lesson, you should be able to answer the following questions regarding sample mean vectors and correlations.

**Sample Mean Vectors**

- What is the distribution of \(\bar{x}\) when data come from a multivariate normal distribution?
- What are the properties of \(\bar{x}\) when data do NOT come from a multivariate normal distribution but the sample size
*n*is large - How to construct a confidence interval for a single multivariate normal population mean
- How to construct confidence intervals for several multivariate normal population means simultaneously

**Sample Correlations**

- How can we test the null hypothesis that there is zero correlation between two variables?
- What can we conclude from such hypothesis tests?
- How can we assess uncertainty regarding estimated correlation coefficients using confidence intervals?
- What is the appropriate interpretation for confidence intervals?