Ch 6 Introduction to Formal Statistical Inference

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Presentation transcript:

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. 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.

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.

For a standard normal curve

Large-sample known s confidence interval for m.

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

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.

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.

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!

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.

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 ?

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.