C HAPTER 11 Section 11.1 – Inference for the Mean of a Population

I NFERENCE FOR THE M EAN OF A P OPULATION

C ONDITIONS FOR I NFERENCE A BOUT A M EAN Our data are a simple random sample (SRS) of size n from the population of interest. This condition is very important. Observations from the population have a normal distribution with mean μ and standard deviation σ. In practice, it is enough that the distribution be symmetric and single-peaked unless the sample is very small. Both μ and σ are unknown parameters.

S TANDARD E RROR

T HE T DISTRIBUTIONS

T DISTRIBUTIONS ( CONTINUED …) The density curves of the t distributions are similar in shape to the standard normal curve. They are symmetric about zero, single-peaked, and bell shaped. The spread of the t distribution is a bit greater than that of the standard normal distribution. The t have more probability in the tails and less in the center than does the standard normal. As the degrees of freedom k increase, the t(k) density curve approached the N(0,1) curve ever more closely.

T HE O NE -S AMPLE T P ROCEDURES

D EGREES OF F REEDOM There is a different t distribution for each sample size. We specify a particular t distribution by giving its degree of freedom. The degree of freedom for the one-sided t statistic come from the sample standard deviation s in the denominator of t. We will write the t distribution with k degrees of freedom as t(k) for short.

E XAMPLE U SING THE “ T T ABLE ”

T HE O NE - SAMPLE T S TATISTIC AND THE T D ISTRIBUTION

T HE O NE - SAMPLE T P ROCEDURE ( CONTINUED …) In terms of a variable T having then t ( n – 1) distribution, the P-value for a test of H o against These P-values are exact if the population distribution is normal and are approximately correct for large n in other cases. H a : μ > μ o is P( T ≥ t) H a : μ < μ o is P( T ≤ t) H a : μ ≠ μ o is 2P( T ≥ |t|)

E XAMPLE A UTO P OLLUTION See example 11.2 on p.622 Minitab stemplot of the data (page 623)

Homework: P.619 #’s 1-4, 8 & 9