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Chapter 11

Chapter 11. Section 11.1 – Inference for the Mean of a Population. Inference for the Mean of a Population.

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Chapter 11

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  1. Chapter 11 Section 11.1 – Inference for the Mean of a Population

  2. 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 . The sampling distribution of 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.

  3. 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.

  4. 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 is .

  5. The t distributions • When we know the value of σ, we base confidence intervals and tests for μ on one-sample z statistics • When we do not know σ, we substitute the standard error of for its standard deviation . • The statistic that results does not have a normal distribution. It has a distribution that is new to us, called a t distribution.

  6. 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.

  7. 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 • Where 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

  8. 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.

  9. 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? • Now suppose you want to construct a 95% confidence interval for the mean of a population based on an SRS of size n = 12. What critical value should you use?

  10. 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 one-sample t statistic has the t distribution with n– 1 degrees of freedom.

  11. 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 • These P-values are exact if the population distribution is normal and are approximately correct for large n in other cases. Ha:μ > μo is P( T ≥ t) Ha:μ < μo is P( T ≤ t) Ha:μ≠ μo is 2P( T ≥|t|)

  12. 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: (where SE stands for “standard error”)

  13. Homework: P.619 #’s 1-4, 8 & 9

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