CHAPTER 11 DAY 1

Assumptions for Inference About a Mean

Our data are a simple random sample (SRS) of size n from the population.

Observations from the population have a normal distribution with mean μand standard deviation σ. Both μand σ are unknown parameters. In the previous chapter we made the

unrealistic assumption that we knew the value of σ, when in practice σ is unknown.

Standard Error

Because we don’t know σ, we estimate it by the sample standard deviation s.

When the standard deviation of a statistic is estimated from the data, the result is called the standard error of the statistic. The standard error of the sample mean is :

xs

n

The One-Sample t Statistic and the t Distributions

Draw an SRS of size n from a population that has the normal distribution with mean μ and standard deviation σ.

The one-sample t statistic

has the t distribution with n – 1 degrees of freedom.

t =x−μ

sn

Facts About t Distributions

The density curves of the t distributions are similar in shape to the standard normal curve. They are symmetric about zero and are bell-shaped.

The spread of the t distributions is a bit greater than that of the standard normal distribution. This comes from using s instead of σ.

As the degrees of freedom increases, the density curve approaches the standard normal curve.

t chart Examples

What critical values from Table C satisfies each of the following conditions?

A. The t distribution with 8 degrees of freedom has probability 0.025 to the right of t*

B. The t distribution with 17 degrees of freedom has probability 0.20 to the left of t*

C. The one-sampled t statistics from a sample of 25 observations has probability 0.01 to the right of t*.

D. The one-sampled t statistics from an SRS of 30 observations has probability 0.95 to the left of t*.

Example

The one-sample t statistic for testing H0: μ= 0 Ha: μ> 0

From a sample of 10 observations has the value t = 3.12 A. What are the degrees of freedom for this statistic? B. Give the two critical values of t* from the Table C

from bracket t. C. Between what two values does the P-value of this

test fall? D. Is the value t = 3.12 significant at the 5% level?

Is it significant at the 1% level?

Confidence Intervals

Confidence interval for t distribution

x ±t*sn

Example

Natalie placed an ad in the newspaper for her beanbags. The following numbers are the beanbag sales from 5 randomly chosen days:37 41 35 3631

Find a 99% confidence interval for the mean number of beanbags sold.