Chapter 11 Day 1

Chapter 11 Day 1Assumptions for Inference About a MeanOur data are a simple random sample (SRS) of size n from the population.

Observations from the population have a normal distribution with mean and standard deviation . Both and are unknown parameters.In the previous chapter we made the unrealistic assumption that we knew the value of , when in practice is unknown.Standard ErrorBecause we dont know , 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 :

The One-Sample t Statistic and the t DistributionsDraw an SRS of size n from a population that has the normal distribution with mean and standard deviation .The one-sample t statistic

has the t distribution with n 1 degrees of freedom.

Facts About t DistributionsThe 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 .As the degrees of freedom increases, the density curve approaches the standard normal curve.

t chart ExamplesWhat 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*.ExampleThe one-sample t statistic for testing H0: = 0Ha: > 0From a sample of 10 observations has the value t = 3.12A. 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 IntervalsConfidence interval for t distribution

ExampleNatalie placed an ad in the newspaper for her beanbags. The following numbers are the beanbag sales from 5 randomly chosen days:3741353631

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