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AP STATISTICS LESSON 11 – 1 (DAY 1)

Learn procedures for making inferences when the population standard deviation is unknown, including t-calculations and procedures for confidence intervals and significance tests.

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AP STATISTICS LESSON 11 – 1 (DAY 1)

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  1. AP STATISTICSLESSON 11 – 1(DAY 1) INFERENCE FOR THE MEAN OF A POPULATION

  2. ESSENTIAL QUESTION: What are the procedures for making inferences when the populations standard deviation is unknown? Objectives: • To become familiar with the procedures for inference when the populations standard deviation is unknown. • To use t calculations and procedures that will lead to making inferences.

  3. Introduction This chapter describes confidence intervals and significance tests for the mean of a single population and for comparing the means of two populations.

  4. 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.

  5. 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.

  6. 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.

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

  8. 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.

  9. 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 t = x –μ has the t distribution with n – 1 degrees of freedom. S/√n

  10. 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.

  11. Example 11.1 Page 619Using 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?

  12. More Practice… What critical value t* from Table C should be used for a confidence interval for the mean of the population in each of the following situations? • A 90% confidence interval based on n = 12 observations? • A 95% confidence interval froman SRS of 30 observations • An 80% confidence interval from a sample of size 18

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