Confidence Intervals for Proportions and Variances

Slides:



Advertisements
Similar presentations
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Chapter 9 Inferences Based on Two Samples.
Advertisements

Chapter 6 Confidence Intervals.
Estimating a Population Variance
Chapter 8 Estimation: Single Population
Chapter 6 Confidence Intervals.
1 (Student’s) T Distribution. 2 Z vs. T Many applications involve making conclusions about an unknown mean . Because a second unknown, , is present,
Confidence Intervals for the Mean (σ Unknown) (Small Samples)
Confidence Intervals Chapter 6. § 6.1 Confidence Intervals for the Mean (Large Samples)
SECTION 6.4 Confidence Intervals for Variance and Standard Deviation Larson/Farber 4th ed 1.
Chapter 6 Confidence Intervals.
Section 7-4 Estimating a Population Mean: σ Not Known.
Chapter 6 Confidence Intervals 1 Larson/Farber 4th ed.
7.2 Confidence Intervals When SD is unknown. The value of , when it is not known, must be estimated by using s, the standard deviation of the sample.
© 2002 Thomson / South-Western Slide 8-1 Chapter 8 Estimation with Single Samples.
6 Chapter Confidence Intervals © 2012 Pearson Education, Inc.
Confidence Intervals Chapter 6. § 6.1 Confidence Intervals for the Mean (Large Samples)
Confidence Intervals Elementary Statistics Larson Farber Chapter 6.
Elementary Statistics
Estimating a Population Variance
Confidence Intervals for Population Proportions
Unit 7 Section : Confidence Intervals for the Mean (σ is unknown)  When the population standard deviation is unknown and our sample is less than.
Confidence Intervals for Population Proportions
Estimating a Population Standard Deviation. Chi-Square Distribution.
381 Hypothesis Testing (Decisions on Means, Proportions and Variances) QSCI 381 – Lecture 28 (Larson and Farber, Sect )
Unit 6 Confidence Intervals If you arrive late (or leave early) please do not announce it to everyone as we get side tracked, instead send me an .
381 Continuous Probability Distributions (The Normal Distribution-II) QSCI 381 – Lecture 17 (Larson and Farber, Sect )
Testing Differences in Population Variances
© Copyright McGraw-Hill 2000
Confidence Intervals for the Mean (Small Samples) 1 Larson/Farber 4th ed.
Confidence Intervals Chapter 6. § 6.3 Confidence Intervals for Population Proportions.
Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall
Estimating a Population Mean. Student’s t-Distribution.
8.1 Estimating µ with large samples Large sample: n > 30 Error of estimate – the magnitude of the difference between the point estimate and the true parameter.
6.4 Confidence Intervals for Variance and Standard Deviation Key Concepts: –Point Estimates for the Population Variance and Standard Deviation –Chi-Square.
Estimating a Population Variance
Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 1 Section 6.2 Confidence Intervals for the Mean (  Unknown)
Section 6.2 Confidence Intervals for the Mean (Small Samples) Larson/Farber 4th ed.
Section 7-5 Estimating a Population Variance. MAIN OBJECTIIVES 1.Given sample values, estimate the population standard deviation σ or the population variance.
Confidence Intervals. Point Estimate u A specific numerical value estimate of a parameter. u The best point estimate for the population mean is the sample.
Section 6.2 Confidence Intervals for the Mean (Small Samples) © 2012 Pearson Education, Inc. All rights reserved. 1 of 83.
Class Seven Turn In: Chapter 18: 32, 34, 36 Chapter 19: 26, 34, 44 Quiz 3 For Class Eight: Chapter 20: 18, 20, 24 Chapter 22: 34, 36 Read Chapters 23 &
Chapter 14 Single-Population Estimation. Population Statistics Population Statistics:  , usually unknown Using Sample Statistics to estimate population.
Chapter 8 Confidence Intervals Copyright © 2014 by The McGraw-Hill Companies, Inc. All rights reserved.McGraw-Hill/Irwin.
CHAPTER 8 Estimating with Confidence
Virtual University of Pakistan
Confidence Intervals and Sample Size
Chapter Eight Estimation.
Chapter 6 Confidence Intervals.
Confidence Intervals and Sample Size
Chapter 6 Confidence Intervals.
Inference for the Mean of a Population
Point and interval estimations of parameters of the normally up-diffused sign. Concept of statistical evaluation.
Chapter 4. Inference about Process Quality
Chapter 6 Confidence Intervals.
Math 4030 – 10b Inferences Concerning Variances: Hypothesis Testing
Estimating a Population Variance
Section 6-4 – Confidence Intervals for the Population Variance and Standard Deviation Estimating Population Parameters.
Estimating Population Variance
M A R I O F. T R I O L A Estimating Population Proportions Section 6-5
STATISTICS INTERVAL ESTIMATION
Chapter 6 Confidence Intervals.
Correlation and Regression-III
Chapter 6 Confidence Intervals.
Confidence Intervals for a Standard Deviation
Section 6-4 – Confidence Intervals for the Population Variance and Standard Deviation Estimating Population Parameters.
Chapter 7 Lecture 3 Section: 7.5.
Elementary Statistics: Picturing The World
Determining Which Method to use
Chapter 6 Confidence Intervals.
Chapter 7 Lecture 3 Section: 7.5.
Presentation transcript:

Confidence Intervals for Proportions and Variances QSCI 381 – Lecture 23 (Larson and Farber, Sects 6.3 and 6.4)

Point Estimate for a Proportion The probability of success in a single trial of a binomial experiment is p. This probability is a population proportion. The point estimate for p, the population proportion of success, is the proportion of successes in the sample, i.e.: Note: is referred to as p-hat.

Example We measure 50 fish, 34 of them have evidence of a parasite. The estimate of the population proportion that have the parasite is:

Confidence Interval for a Proportion We can calculate approximate confidence intervals for an estimate of a proportion using the normal approximation to the binomial distribution, i.e.: A for the population proportion p is: where: c-confidence interval

Example Find a 90% confidence interval for the proportion of our fish population that has the parasite: Identify n, and . Check whether the binomial distribution can be approximated by a normal distribution, i.e. Determine the critical value . Calculate the maximum error of estimate. Construct the c-confidence interval.

Minimum Sample Size to Estimate p Given a c-confidence level and a maximum error of estimate E, the minimum sample size n needed to estimate p is: This formula depends on and which are the quantities were are trying to estimate. Either set the values for these quantities to preliminary estimates or set . Why is this latter assumption “conservative”?

Example You wish to sample a population and you want to estimate, with 95% confidence, the proportion that are mature to within 0.01. How large must the sample size be?

Point Estimate for a Variance The point estimate for 2 is s2 and the point estimate for  is s. s2 is the most unbiased estimate of 2.

The Chi-square Distribution-I If the random variable X has a normal distribution, then the distribution of forms a for samples of any size n>1. chi-square distribution

The Chi-square Distribution-II The properties of the chi-square distribution are: All values of 2 are greater than or equal to zero. The area under the chi-square distribution equals one. Chi-square distributions are positively skewed. The chi-square distribution is a family of curves, each determined by the degrees of freedom. To form a confidence interval for 2, use the chi-square distribution with degrees of freedom equal to one less than the sample size, i.e. d.f.=n-1.

The Chi-square Distribution-III

The Chi-square Distribution-IV with 10 degrees of freedom Note: the distribution is not symmetric. (1-c)/2

The Chi-square Distribution-V CHIINV(p,d.f.) Chiinv(0.05,10) = 18.307 Chiinv(0.95,10) = 3.940

Confidence Intervals for 2 and  A c-confidence interval for a population variance and standard deviation is:

Example The density of a fish species is estimated by taking 25 samples. The sample standard deviation is 10 kg / ha. Construct a 95% confidence interval for the population standard deviation. We first find the critical chi-square values. We want a 95% confidence interval so the probability below the left limit and above the right limit should be 0.025. Note that the d.f. is 24 (n=25) We can now construct the confidence interval: or