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Chapter 11 Day 1

Chapter 11 Day 1. Assumptions for Inference About a Mean. Our data are a simple random sample (SRS) of size n from the population. Observations from the population have a normal distribution with mean μand standard deviation σ . Both μand σ are unknown parameters.

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Chapter 11 Day 1

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  1. Chapter 11 Day 1

  2. Assumptions for Inference About a Mean • Our data are a simple random sample (SRS) of size n from the population. • Observations from the population have a normal distribution with mean μand standard deviation σ. Both μandσ are unknown parameters. • In the previous chapter we made the unrealistic assumption that we knew the value of σ, when in practice σ is unknown.

  3. Standard Error • Because we don’t know σ, we estimate it by the sample standard deviation s. • 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 :

  4. The One-Sample t Statistic and the tDistributions • 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 tdistribution with n – 1 degrees of freedom.

  5. Facts About t Distributions • The density curves of the t distributions are similar in shape to the standard normal curve. They are symmetric about zero and are bell-shaped. • The spread of the t distributions is a bit greater than that of the standard normal distribution. This comes from using s instead of σ. • As the degrees of freedom increases, the density curve approaches the standard normal curve.

  6. t chart Examples • What critical values from Table C satisfies each of the following conditions? • A. The t distribution with 8 degrees of freedom has probability 0.025 to the right of t* • B. The tdistribution with 17 degrees of freedom has probability 0.20 to the left of t*

  7. C. The one-sampled t statistics from a sample of 25 observations has probability 0.01 to the right of t*. • D. The one-sampled t statistics from an SRS of 30 observations has probability 0.95 to the left of t*.

  8. Example • The one-sample t statistic for testing • H0: μ= 0 • Ha: μ> 0 From a sample of 10 observations has the value t = 3.12 • A. What are the degrees of freedom for this statistic? • B. Give the two critical values of t* from the Table C from bracket t. • C. Between what two values does the P-value of this test fall? • D. Is the value t = 3.12 significant at the 5% level? Is it significant at the 1% level?

  9. Confidence Intervals • Confidence interval for tdistribution

  10. Example • Natalie placed an ad in the newspaper for her beanbags. The following numbers are the beanbag sales from 5 randomly chosen days: 37 41 35 36 31 • Find a 99% confidence interval for the mean number of beanbags sold.

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