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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. In the previous chapter we made the unrealistic assumption that we knew the value of σ, when in practice σ is unknown. 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 x is : s n The One-Sample t Statistic and the t Distributions 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 x t s n has the t distribution with n – 1 degrees of freedom. 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. 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 t distribution with 17 degrees of freedom has probability 0.20 to the left of t* 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*. 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? Confidence Intervals Confidence interval for t distribution s xt* n 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.