Understanding Confidence Intervals: Basics and Calculation, Study notes of Statistics

Download Understanding Confidence Intervals: Basics and Calculation and more Study notes Statistics in PDF only on Docsity! Confidence intervals: The basics BPS chapter 14 © 2006 W.H. Freeman and Company Objectives (BPS chapter 14) Confidence intervals: the basics Estimating with confidence Confidence intervals for the mean μ How confidence intervals behave Choosing the sample size Red dot: mean value of individual sample 95% of all sample means will be within roughly 2 standard deviations (2*σ/√n) of the population parameter μ. Because distances are symmetrical, this implies that the population parameter μ must be within roughly 2 standard deviations from the sample average , in 95% of all samples. σ n This reasoning is the essence of statistical inference. x The weight of single eggs of the brown variety is normally distributed N(65g,5g). Think of a carton of 12 brown eggs as an SRS of size 12. You buy a carton of 12 white eggs instead. The box weighs 770g. The average egg weight from that SRS is thus = 64.2g. Knowing that the standard deviation of egg weight is 5g, what can you infer about the mean µ of the white egg population? There is a 95% chance that the population mean µ is roughly within ± 2σ/√n of , or 64.2g ± 2.88g. population sample What is the distribution of the sample means ? Normal (mean μ, standard deviation σ/√n) = N(65g,1.44g). Find the middle 95% of the sample means distribution. Roughly ± 2 standard deviations from the mean, or 65g ± 2.88g. x x x Confidence interval A level C confidence interval for a parameter has two parts: An interval calculated from the data, usually of the form estimate ±margin of error A confidence level C, which gives the probability that the interval will capture the true parameter value in repeated samples, or the success rate for the method. Interpreting a confidence interval for a mean A confidence interval can be expressed as: ± m m is called the margin of error x Two endpoints of an interval: μ possibly within ( − m) to ( + m) Example: 114 to 126 A confidence level C (in %) indicates the success rate of the method that produces the interval. It represents the area under the normal curve within ± m of the center of the curve. mm x x x x ?μ Review: standardizing the normal curve using z N(0,1) zx N(64.5, 2.5) N(µ, σ/√n) Standardized height (no units) z = x − μ σ n Here, we work with the sampling distribution, and σ/√n is its standard deviation (spread). Remember that σ is the standard deviation of the original population. Confidence intervals contain the population mean μ in C% of samples. Different areas under the curve give different confidence levels C. Example: For an 80% confidence level C, 80% of the normal curve’s area is contained in the interval. C Z*−Z* Varying confidence levels Practical use of z: z* z* is related to the chosen confidence level C. C is the area under the standard normal curve between −z* and z*. x ± z *σ n The confidence interval is thus: Different confidence intervals for the same set of measurements 96% confidence interval for the true density, z* = 2.054, and write = 28 ± 2.054(1/√3) = 28 ± 1.19*106 bacteria/ml 70% confidence interval for the true density, z* = 1.036, and write = 28 ± 1.036(1/√3) = 28 ± 0.60*106 bacteria/ml Density of bacteria in solution: Measurement equipment has standard deviation σ = 1*106 bacteria/ml fluid. 3 measurements: 24, 29, and 31*106 bacteria/ml fluid Mean: = 28*106 bacteria/ml. Find the 96% and 70% CI. n zx σ*± n zx σ*± x Impact of sample size The spread in the sampling distribution of the mean is a function of the number of individuals per sample. The larger the sample size, the smaller the standard deviation (spread) of the sample mean distribution. But the spread only decreases at a rate equal to √n. Sample size n S ta nd ar d er ro r σ ⁄√ n Sample size and experimental design You may need a certain margin of error (e.g., drug trial, manufacturing specs). In many cases, the population variability (σ) is fixed, but we can choose the number of measurements (n). So plan ahead what sample size to use to achieve that margin of error. m = z * σ n ⇔ n = z *σ m ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 2 Remember, though, that sample size is not always stretchable at will. There are typically costs and constraints associated with large samples. The best approach is to use the smallest sample size that can give you useful results.