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Section 11.1 Part 1 β Inference for the Mean of a Population Inference for the Mean of a Population ο· Confidence intervals and tests of significance for the mean ΞΌ of a normal population are based on the sample mean π₯Μ . ο· The sampling distribution of π₯Μ has ΞΌ as its mean. ο· ο· That is an __________________________________ of the unknown ΞΌ. In the previous chapter we make the unrealistic assumption that we knew the value of Ο. In practice, _____ ___________________. Conditions for Inference About a Mean ο· Our data are a ______________________________________________ from the population of interest. This condition is very important. ο· Observations from the population have a______________________________________________________. In practice, it is enough that the distribution be ___________________________________________ unless the ________________________________. ο· Both ΞΌ and Ο are unknown parameters. Standard Error ο· When the standard deviation of a statistic is ____________________________________, the result is called the ________________________________of the statistic. ο· The standard error of the sample mean π₯Μ is _________________________ The t distributions ο· When we know the value of Ο, we base confidence intervals and tests for ΞΌ on _______________________ π§= ο· π₯Μ β π π βπ When we do not know Ο, we substitute the ____________________ of π₯Μ for its standard deviation _______ ο· The statistic that results does not have a normal distribution. It has a distribution that is new to us, called a ___________________________. ο· The density curves of the t distributions are _________________________________to the standard normal curve. They are_______________________________________________________________________ ο· The spread of the t distribution is a bit _______________________ that of the standard normal distribution. The t have ___________________________in the ___________________and ___________ in the __________________________ than does the standard normal. ο· As the _______________________________________________________, the t(k) density curve approaches the ____________________ curve ever more closely. The One-sample t Statistic and the t Distribution ο· Draw an SRS of size n from a population that has the normal distribution with mean ΞΌ and standard deviation Ο. ________________________________________has the t distribution with _______________ _______________________________. π‘= π₯Μ β π π βπ Degrees of Freedom ο· There is a different t distribution for each sample size. We specify a particular t distribution by giving its degree of freedom. ο· The degree of freedom for the one-sided t statistic come from the sample standard deviation s in the denominator of t. ο· We will write the t distribution with k degrees of freedom as ____________ for short. Example 11.1 - Using the βt Tableβ ο· What critical value t* from Table C (back cover of text book, often referred to as the βt tableβ) would you use for a t distribution with 18 degrees of freedom having probability 0.90 to the left of t? ο ο· π‘ β = 1.330 Now suppose you want to construct a 95% confidence interval for the mean π of a population based on an SRS of size n = 12. What critical value π‘ β should you use? ο π‘ β = 2.201 The One-Sample t Procedures ο· Draw an SRS of size n from a population having unknown mean ΞΌ. A level C confidence interval for ΞΌ is π₯Μ ± π‘ β π βπ ο· Where π‘ β is the upper _____________ critical value for the _________________distribution. This interval is exact when the population distribution is normal and is approximately correct for large n in other cases. ο· The test the hypothesis H0 : ΞΌ = ΞΌ0 based on an SRS of size n, computed the _________________________. ο· In terms of a variable T having then t(n β 1) distribution, the P-value for a test of Ho against ο· These P-values are exact if the population distribution is normal and are approximately correct for large n in other cases. Example 11.2 - Auto Pollution ο· See example 11.2 on p.622 ο· The one-sample t confidence interval has the form: estimate ± π‘ β SEestimate (where SE stands for βstandard errorβ)