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Ch 6 Introduction to Formal Statistical Inference

Ch 6 Introduction to Formal Statistical Inference. 6.1 Large Sample Confidence Intervals for a Mean. A confidence interval for a parameter is a data-based interval of numbers likely to include the true value of the parameter with a probability-based confidence.

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Ch 6 Introduction to Formal Statistical Inference

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  1. Ch 6 Introduction to Formal Statistical Inference

  2. 6.1 Large Sample Confidence Intervals for a Mean • A confidence interval for a parameter is a data-based interval of numbers likely to include the true value of the parameter with a probability-based confidence. • A 95% confidence interval for µ is an interval which was constructed in a manner such that 95% of such intervals contain the true value of µ.

  3. Interval Estimate—Confidence intervals • An interval estimate consists of an interval which will contain the quantity it is supposed to estimate with a specified probability (or degree of confidence). • Recall that for large random samples, the sampling distribution of the mean is approximately a normal distribution with • So we will utilize some properties of normal distribution to explain a confidence interval.

  4. For a standard normal curve

  5. Large-sample known s confidence interval for m.

  6. Confidence Intervals 100(1-a)% CI: 80% 90% 95% 99%

  7. Confidence Interval for Means After computing sample mean , find a range of values such that 95% of the time the resulting range includes the true value m.

  8. X=breaking strength of a fish line. σ=0.10. In a random sample of size n=10, Find a 95% confidence interval for μ, the true average breaking strength.

  9. How large a sample size is needed in order to get an error of no more than 0.01 with 95% probability if the sample mean is used to estimate the true mean? • Solution n=385, always round up!

  10. Example • A certain adjustment to a machine will change the length of the parts it is making but will not affect the standard deviation. • The length of the parts is normally distributed, and the standard deviation is 0.5 mm (millimeter). • After an adjustment is made, a random sample is taken to determine the mean length of parts now being produced. The observed lengths are 75.3, 76.0, 75.0, 77.0, 75.4, 76.3, 77.0, 74.9, 76.5, 75.8.

  11. Questions • What is the parameter of interest? • Find the point estimate of the mean length of parts now being produced. c. Find the 99% confidence interval for μ. d. How large a sample should be taken if the population mean is to be estimated with 99% confidence to have an error not exceeding 0.2 mm ?

  12. Solution a.The mean length of parts now being produced (μ); b. x=75.92 c. n=10; σ=0.5; . The 99% confidence interval is 75.512<μ<76.328 • Δ=0.20; since n must be an integer, n=42.

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