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CHAPTER 11 Section 11.1 – Inference for the Mean of a Population INFERENCE FOR THE MEAN OF A POPULATION Confidence intervals and tests of significance for the mean μ of a normal population are based on the sample mean 𝑥. The sampling distribution of 𝑥 has μ as its mean. That is an unbiased estimator of the unknown μ. In the previous chapter we make the unrealistic assumption that we knew the value of σ. In practice, σ is unknown. CONDITIONS FOR INFERENCE ABOUT A MEAN 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. STANDARD ERROR 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 𝑥 is 𝑠 . 𝑛 THE T DISTRIBUTIONS When we know the value of σ, we base confidence intervals and tests for μ on one-sample z statistics 𝑥−𝜇 𝑧= 𝜎 𝑛 When 𝑠we do not know σ, we substitute 𝜎the standard error 𝑛 of 𝑥 for its standard deviation 𝑛 . The statistic that results does not have a normal distribution. It has a distribution that is new to us, called a t distribution. 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. THE ONE-SAMPLE T PROCEDURES Draw an SRS of size n from a population having unknown mean μ. A level C confidence interval for μ is 𝑠 𝑥 ± 𝑡∗ 𝑛 Where 𝑡 ∗ is the upper (1 - C)/2 critical value for the t(n – 1) distribution. This interval is exact when the population distribution is normal and is approximately correct for large n in other cases. The test the hypothesis H0 : μ = μ0 based on an SRS of size n, computed the one-sample t statistic 𝑥−𝜇 𝑡= 𝑠0 𝑛 DEGREES OF FREEDOM 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. EXAMPLE 11.1 - USING THE “T TABLE” What critical value t* from Table C (back cover of text book, often referred to as the “t table”) would you use for a t distribution with 18 degrees of freedom having probability 0.90 to the left of t? 𝑡 ∗ = 1.330 Now suppose you want to construct a 95% confidence interval for the mean 𝜇 of a population based on an SRS of size n = 12. What critical value 𝑡 ∗ should you use? 𝑡 ∗ = 2.201 THE ONE-SAMPLE T STATISTIC AND THE T DISTRIBUTION Draw an SRS of size n from a population that has the normal distribution with mean μ and standard deviation σ. The onesample t statistic has the t distribution with n – 1 degrees of freedom. 𝑥−𝜇 𝑡= 𝑠 𝑛 THE ONE-SAMPLE T PROCEDURE (CONTINUED…) In terms of a variable T having then t(n – 1) distribution, the P-value for a test of Ho against Ha: μ > μo is P( T ≥ t) Ha: μ < μo is P( T ≤ t) Ha: μ ≠ μo is 2P( T ≥ |t|) 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 See example 11.2 on p.622 Minitab stemplot of the data (page 623) The one-sample t confidence interval has the form: estimate ± 𝑡 ∗ SEestimate (where SE stands for “standard error”) Homework: P.619 #’s 1-4, 8 & 9