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 μ and standard deviation σ. Both μ and 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 :

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.

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 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  H 0 : μ = 0  H a : 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

Example  Natalie placed an ad in the newspaper for her beanbags. The following numbers are the beanbag sales from 5 randomly chosen days:  Find a 99% confidence interval for the mean number of beanbags sold.