April 18, 2009 - Class Notes (2nd Half of Class)

“Until now, you have focused on the first branch of statistics—descriptive statistics and probability.
You have learned to describe and graph data, calculate probabilities, and use properties of normal
distributions. In this unit, you will learn about inferential statistics.
This chapter will focus on how to estimate a population parameter and state how confident you are
about your estimate.”

* * * * * * *
A. Confidence Intervals for the Mean (Large Samples)

A point estimate is a single value estimate for a population parameter.
An interval estimate is an interval, or range of values, that is used to estimate the population
parameter.

The level of confidence c is the probability that the interval estimate contains the population
parameter. It states how confident we are that the interval estimate contains the population
parameter.

The difference between the point estimate and the actual population parameter value is called the sampling error and is denoted by x − μ .
Given a level of confidence, the greatest possible sampling error is called the margin of error. It is, sometimes, also called the maximum error of estimate or error tolerance and is denoted by E.
When the population standard deviation is known, E can be calculated using the formula:

n
E zc x zc
σ
= σ = .
A c-confidence interval for the population mean μ is written as:
x − E < μ < x + E

Here, c is the probability that the confidence interval contains μ.
The steps for determining the confidence interval for a population mean, when the sample size
n ≥ 30 or the sample comes from a normally distributed population, can be listed as follows:

Step 1: Find the sample statistics.
Step 2: Calculate standard deviation, s.
Step 3: Find the critical values.
Step 4: Calculate the margin of error.
Step 5: Form the confidence interval x − E < μ < x + E .

B. Confidence Intervals for the Mean (Small Samples)

A t-distribution is used when a sample size is less than 30, and the random variable x is
approximately normally distributed. The properties of t-distribution are as follows:

1. It is bell-shaped and symmetric about the mean.
2. The mean, median, and mode of the t-distribution are equal to zero.
3. The total area under a t-curve is 1, or 100%.
4. The t-distribution is a family of curves, each determined by a parameter called the degrees
of freedom, also referred to as d.f. The degrees of freedom are the number of free choices
left after a sample statistic such as x is calculated. When you use a t-distribution to estimate
a population mean, the degrees of freedom are equal to one less than the sample size.
d.f. = n – 1
5. As the degrees of freedom increase, the t-distribution approaches the normal distribution.
After 30 degrees of freedom, the t-distribution is very close to the standard normal z distribution.

Constructing a confidence interval using the t-distribution involves using a point estimate and a
margin of error. The following steps can be used for constructing a confidence interval for the mean of t-distribution.

Step 1: Assuming that the sample comes from a normally distributed population, identify the
sample statistics n, x , and variance s. Use the formulas
n
x
x Σ = and
1
( )2
−
−
= Σ
n
x x
s .

Step 2: Identify the degrees of freedom, the level of confidence c, and the critical value tc using the t-distribution table. Remember, d.f. = n – 1.

Step 3: Find the margin of error E using the formula
n
s
E = tc .

Step 4: Find the left and right endpoints and form the confidence interval. Use the following
formulas:
• Left endpoints: x − E
• Right endpoints: x + E
• Interval: x − E < μ < x + E