INFERENTIAL STATISTICS REVISION
|
|
|
- Scott Evans
- 5 years ago
- Views:
Transcription
1 INFERENTIAL STATISTICS REVISION
2 PREMIUM VERSION PREVIEW
3 2016 LCHL Paper 2 Question 9 (a) (i) Data on earnings were published for a particular country. The data showed that the annual income of people in full-time employment was normally distributed with a mean of and a standard deviation of The government intends to impose a new tax on incomes over Find the percentage of full-time workers who will be liable for this tax, correct to one decimal place. Calculate the z score. x μ σ x is a data point μ is the population mean σ is the population standard deviation Quick Sketch 39, μ = 39,400 σ = 12,920 x = 60,000 60,000 The tables only give values to the left of z but: P Z 1 P Z z P z > 1.59 = 1 P z < 1.59 = = %
4 2016 LCHL Paper 2 Question 9 (a) (ii) The government will also provide a subsidy to the lowest 10 % of income earners. Find the level of income at which the government will stop paying the subsidy, correct to the nearest euro. 39,400 Instead of calculating the lowest 10% we calculate those above 90% as the curve is symmetrical and the tables have no percentages less than 50% = x μ σ = 1.28 Find the z score that corresponds to 0.90 x μ σ 0.10 Quick Sketch x = x = x = Less than 90% is the z score 1.28 BUT greater than 90% is 1.28 (same as less than 10%). x = 22,862
5 2016 LCHL Paper 2 Question 9 (a) (iii) Some time later a research institute surveyed a sample of 1000 full-time workers, randomly selected, and found that the mean annual income of the sample was Test the hypothesis, at the 5% level of significance, that the mean annual income of full-time workers has changed since the national data were published. State the null hypothesis and the alternative hypothesis. Give your conclusion in the context of the question. 15 marks 11 for the z score. Was acceptable to use Confidence Interval Method State the null and alternate hypothesis H 0 : μ = 39400, the mean annual income of full time workers is still 39,400 H 0 : μ 39400, the mean annual income of full time workers is NOT still 39,400 Calculate the test statistic x μ σ n x = 38,280 μ = 39,400 σ = 12,920 n = 1, < 1.96 Reject the null hypothesis. Sketch of Critical Regions Give conclusion in the context of the question. There is enough evidence to reject the claim that the mean annual income has remained the same. Marking scheme: The mean income has changed.
6 2016 LCHL Paper 2 Question 9 (b) The research institute surveyed 400 full-time farmers, randomly selected from all the full-time farmers in the country, and found that the mean income for the sample was and the standard deviation was Assuming that annual farm income is normally distributed in this country, create a 95% confidence interval for the mean income of full-time farmers. 10 marks 95% Confidence Interval for a Means x 1.96 σ n < μ < x σ n < μ < < μ < < μ < Margin of Error (Mean) ±1.96 σ n Note: σ is in the tables but you must n remember the ±1.96 which is specific to a 95% level of confidence.
7 2016 LCHL Paper 2 Question 9 (c) It is known that data on farm size are not normally distributed. The research institute could take many large random samples of farm size and create a sampling distribution of the means of all these samples. Give one reason why they might do this. Central Limit Theorem Mean μ x = μ Standard Deviation σ x = σ n The distribution of sample means will be normally distributed. The mean of the samples will be equal to the mean of the population (for n > 30) and the standard deviation will be σ n. Must know the above!
8 2016 LCHL Paper 2 Question 9 (d) The research institute also carried out a survey into the use of agricultural land. n farmers were surveyed. If the margin of error of the survey was 4 5 %, find the value of n. Without more information this is the only formula we could use. Syllabus probably needs a little clarity here. Margin or Error 1 n 1 n = = n n = The institute interviewed 494 farmers.
9 2015 LCHL Paper 2 Question 2 (a) A survey of 100 shoppers, randomly selected from a large number of Saturday supermarket shoppers, showed that the mean shopping spend was The standard deviation of this sample was Find a 95% confidence interval for the mean amount spent in a supermarket on that Saturday. 10 marks Confidence Interval for a Mean x 1.96 σ n < μ < x σ n < μ < < μ < < μ < Margin of Error (Mean) ±1.96 σ n Note: σ is in the tables but you must n remember the ±1.96 which is specific to a 95% level of confidence. We can be 95% confident that the mean amount spent was in the range < μ < 94.51
10 2015 LCHL Paper 2 Question 2 (b) A supermarket has claimed that the mean amount spent by shoppers on a Saturday is 94. Based on the survey, test the supermarket s claim using a 5% level of significance. Clearly state your null hypothesis, your alternative hypothesis, and your conclusion. Full marks for Confidence Interval Method EVEN if you already used it for part (a). That would be 20 marks for doing the same thing twice!! State the null and alternate hypothesis H 0 : μ = 94, the mean amount spent by shoppers on a Saturday is 94 H 1 : μ 94, the mean amount spent by shoppers on a Saturday is NOT 94 Calculate the test statistic x μ σ n x = μ = 90 σ = n = > 1.96 Fail to reject H 0 Sketch of Critical Regions Give conclusion in the context of the question. We do not reject the null hypothesis as 1.71 is in the fail to reject region. This means that there is not sufficient evidence to reject the claim that the mean amount spent by shoppers is as the supermarket claims. Marking scheme: Outlined no conclusion.
11 2015 LCHL Paper 2 Question 2 (c) Find the p value of the test you performed in part (b) above and explain what this value represents in the context of the question. p value = 2 P z > z 1 where z 1 is the test statistic The probability of getting a value z < 1.71 is got from the tables. P z < 1.71 = 1 P z < 1.71 = = The probability of getting a value > 1.71 is also p value is the sum of these probabilities = Compare this to the level of significance 5% or > 0.05 This is greater evidence to fail to reject the null hypothesis. What does the p value represent: If the supermarkets claim is true then our sample result would be this far (a measure of = 3.55) from the true claim 8.72% of the time. We are unwilling to reject it as this % is too high (i.e. greater than 5%). p-value is easy to calculate but interpreting it has caused difficulty.
12 PREMIUM VERSION PREVIEW
Making Sense of Cents
Name: Date: Making Sense of Cents Exploring the Central Limit Theorem Many of the variables that you have studied so far in this class have had a normal distribution. You have used a table of the normal
The Normal Probability Distribution
1 The Normal Probability Distribution Key Definitions Probability Density Function: An equation used to compute probabilities for continuous random variables where the output value is greater than zero
Section Introduction to Normal Distributions
Section 6.1-6.2 Introduction to Normal Distributions 2012 Pearson Education, Inc. All rights reserved. 1 of 105 Section 6.1-6.2 Objectives Interpret graphs of normal probability distributions Find areas
Mr. Orchard s Math 141 WIR 8.5, 8.6, 5.1 Week 13
1. Find the following probabilities, where Z is a random variable with a standard normal distribution and X is a normal random variable with mean µ = 380 and standard deviation σ = 21: (Round your answers
Section 6.5. The Central Limit Theorem
Section 6.5 The Central Limit Theorem Idea Will allow us to combine the theory from 6.4 (sampling distribution idea) with our central limit theorem and that will allow us the do hypothesis testing in the
Mean GMM. Standard error
Table 1 Simple Wavelet Analysis for stocks in the S&P 500 Index as of December 31 st 1998 ^ Shapiro- GMM Normality 6 0.9664 0.00281 11.36 4.14 55 7 0.9790 0.00300 56.58 31.69 45 8 0.9689 0.00319 403.49
Copyright 2005 Pearson Education, Inc. Slide 6-1
Copyright 2005 Pearson Education, Inc. Slide 6-1 Chapter 6 Copyright 2005 Pearson Education, Inc. Measures of Center in a Distribution 6-A The mean is what we most commonly call the average value. It is
The Normal Distribution
5.1 Introduction to Normal Distributions and the Standard Normal Distribution Section Learning objectives: 1. How to interpret graphs of normal probability distributions 2. How to find areas under the
MA131 Lecture 8.2. The normal distribution curve can be considered as a probability distribution curve for normally distributed variables.
Normal distribution curve as probability distribution curve The normal distribution curve can be considered as a probability distribution curve for normally distributed variables. The area under the normal
Estimation of the Mean and Proportion
Chapter 8 Estimation of the Mean and Proportion In statistics, we collect samples to know more about a population. If the sample is representative of the population, the sample mean or proportion should
Chapter Seven: Confidence Intervals and Sample Size
Chapter Seven: Confidence Intervals and Sample Size A point estimate is: The best point estimate of the population mean µ is the sample mean X. Three Properties of a Good Estimator 1. Unbiased 2. Consistent
Central Limit Theorem
Central Limit Theorem Lots of Samples 1 Homework Read Sec 6-5. Discussion Question pg 329 Do Ex 6-5 8-15 2 Objective Use the Central Limit Theorem to solve problems involving sample means 3 Sample Means
Key Objectives. Module 2: The Logic of Statistical Inference. Z-scores. SGSB Workshop: Using Statistical Data to Make Decisions
SGSB Workshop: Using Statistical Data to Make Decisions Module 2: The Logic of Statistical Inference Dr. Tom Ilvento January 2006 Dr. Mugdim Pašić Key Objectives Understand the logic of statistical inference
Normal Probability Distributions
Normal Probability Distributions Properties of Normal Distributions The most important probability distribution in statistics is the normal distribution. Normal curve A normal distribution is a continuous
Chapter 7. Sampling Distributions
Chapter 7 Sampling Distributions Section 7.1 Sampling Distributions and the Central Limit Theorem Sampling Distributions Sampling distribution The probability distribution of a sample statistic. Formed
In a binomial experiment of n trials, where p = probability of success and q = probability of failure. mean variance standard deviation
Name In a binomial experiment of n trials, where p = probability of success and q = probability of failure mean variance standard deviation µ = n p σ = n p q σ = n p q Notation X ~ B(n, p) The probability
Determining Sample Size. Slide 1 ˆ ˆ. p q n E = z α / 2. (solve for n by algebra) n = E 2
Determining Sample Size Slide 1 E = z α / 2 ˆ ˆ p q n (solve for n by algebra) n = ( zα α / 2) 2 p ˆ qˆ E 2 Sample Size for Estimating Proportion p When an estimate of ˆp is known: Slide 2 n = ˆ ˆ ( )
Lecture Slides. Elementary Statistics Twelfth Edition. by Mario F. Triola. and the Triola Statistics Series. Section 7.4-1
Lecture Slides Elementary Statistics Twelfth Edition and the Triola Statistics Series by Mario F. Triola Section 7.4-1 Chapter 7 Estimates and Sample Sizes 7-1 Review and Preview 7- Estimating a Population
Lecture Slides. Elementary Statistics Tenth Edition. by Mario F. Triola. and the Triola Statistics Series. Slide 1
Lecture Slides Elementary Statistics Tenth Edition and the Triola Statistics Series by Mario F. Triola Slide 1 Chapter 6 Normal Probability Distributions 6-1 Overview 6-2 The Standard Normal Distribution
7.1 Comparing Two Population Means: Independent Sampling
University of California, Davis Department of Statistics Summer Session II Statistics 13 September 4, 01 Lecture 7: Comparing Population Means Date of latest update: August 9 7.1 Comparing Two Population
19. CONFIDENCE INTERVALS FOR THE MEAN; KNOWN VARIANCE
19. CONFIDENCE INTERVALS FOR THE MEAN; KNOWN VARIANCE We assume here that the population variance σ 2 is known. This is an unrealistic assumption, but it allows us to give a simplified presentation which
A continuous random variable is one that can theoretically take on any value on some line interval. We use f ( x)
Section 6-2 I. Continuous Probability Distributions A continuous random variable is one that can theoretically take on any value on some line interval. We use f ( x) to represent a probability density
Statistics for Managers Using Microsoft Excel 7 th Edition
Statistics for Managers Using Microsoft Excel 7 th Edition Chapter 7 Sampling Distributions Statistics for Managers Using Microsoft Excel 7e Copyright 2014 Pearson Education, Inc. Chap 7-1 Learning Objectives
7. For the table that follows, answer the following questions: x y 1-1/4 2-1/2 3-3/4 4
7. For the table that follows, answer the following questions: x y 1-1/4 2-1/2 3-3/4 4 - Would the correlation between x and y in the table above be positive or negative? The correlation is negative. -
6.1 Graphs of Normal Probability Distributions:
6.1 Graphs of Normal Probability Distributions: Normal Distribution one of the most important examples of a continuous probability distribution, studied by Abraham de Moivre (1667 1754) and Carl Friedrich
6 Central Limit Theorem. (Chs 6.4, 6.5)
6 Central Limit Theorem (Chs 6.4, 6.5) Motivating Example In the next few weeks, we will be focusing on making statistical inference about the true mean of a population by using sample datasets. Examples?
Chapter Seven. The Normal Distribution
Chapter Seven The Normal Distribution 7-1 Introduction Many continuous variables have distributions that are bellshaped and are called approximately normally distributed variables, such as the heights
Section 3.4 The Normal Distribution
Section 3.4 The Normal Distribution Properties of the Normal Distribution Curve 1. We denote the normal random variable with X = x. 2. The curve has a peak at x = µ. 3. The curve is symmetric about the
Chapter 5 Basic Probability
Chapter 5 Basic Probability Probability is determining the probability that a particular event will occur. Probability of occurrence = / T where = the number of ways in which a particular event occurs
Chapter 6 Confidence Intervals
Chapter 6 Confidence Intervals Section 6-1 Confidence Intervals for the Mean (Large Samples) VOCABULARY: Point Estimate A value for a parameter. The most point estimate of the population parameter is the
Chapter 6: The Normal Distribution
Chapter 6: The Normal Distribution Diana Pell Section 6.1: Normal Distributions Note: Recall that a continuous variable can assume all values between any two given values of the variables. Many continuous
8.3 CI for μ, σ NOT known (old 8.4)
GOALS: 1. Learn the properties of the student t distribution and the t curve. 2. Understand how degrees of freedom, df, relate to t curves. 3. Recognize that t curves approach the SNC as df increases.
C.10 Exercises. Y* =!1 + Yz
C.10 Exercises C.I Suppose Y I, Y,, Y N is a random sample from a population with mean fj. and variance 0'. Rather than using all N observations consider an easy estimator of fj. that uses only the first
NORMAL RANDOM VARIABLES (Normal or gaussian distribution)
NORMAL RANDOM VARIABLES (Normal or gaussian distribution) Many variables, as pregnancy lengths, foot sizes etc.. exhibit a normal distribution. The shape of the distribution is a symmetric bell shape.
Version A. Problem 1. Let X be the continuous random variable defined by the following pdf: 1 x/2 when 0 x 2, f(x) = 0 otherwise.
Math 224 Q Exam 3A Fall 217 Tues Dec 12 Version A Problem 1. Let X be the continuous random variable defined by the following pdf: { 1 x/2 when x 2, f(x) otherwise. (a) Compute the mean µ E[X]. E[X] x
Lecture 8: Single Sample t test
Lecture 8: Single Sample t test Review: single sample z-test Compares the sample (after treatment) to the population (before treatment) You HAVE to know the populational mean & standard deviation to use
Confidence Interval and Hypothesis Testing: Exercises and Solutions
Confidence Interval and Hypothesis Testing: Exercises and Solutions You can use the graphical representation of the normal distribution to solve the problems. Exercise 1: Confidence Interval A sample of
AMS7: WEEK 4. CLASS 3
AMS7: WEEK 4. CLASS 3 Sampling distributions and estimators. Central Limit Theorem Normal Approximation to the Binomial Distribution Friday April 24th, 2015 Sampling distributions and estimators REMEMBER:
Midterm Test 1 (Sample) Student Name (PRINT):... Student Signature:... Use pencil, so that you can erase and rewrite if necessary.
MA 180/418 Midterm Test 1 (Sample) Student Name (PRINT):............................................. Student Signature:................................................... Use pencil, so that you can erase
ECON 214 Elements of Statistics for Economists
ECON 214 Elements of Statistics for Economists Session 7 The Normal Distribution Part 1 Lecturer: Dr. Bernardin Senadza, Dept. of Economics Contact Information: bsenadza@ug.edu.gh College of Education
A LEVEL MATHEMATICS QUESTIONBANKS NORMAL DISTRIBUTION - BASIC
1. The random variable X has a normal distribution with mean 5 and standard deviation 2. Find: a) P(X
(# of die rolls that satisfy the criteria) (# of possible die rolls)
BMI 713: Computational Statistics for Biomedical Sciences Assignment 2 1 Random variables and distributions 1. Assume that a die is fair, i.e. if the die is rolled once, the probability of getting each
Chapter 7 Study Guide: The Central Limit Theorem
Chapter 7 Study Guide: The Central Limit Theorem Introduction Why are we so concerned with means? Two reasons are that they give us a middle ground for comparison and they are easy to calculate. In this
STATISTICS - CLUTCH CH.9: SAMPLING DISTRIBUTIONS: MEAN.
!! www.clutchprep.com SAMPLING DISTRIBUTIONS (MEANS) As of now, the normal distributions we have worked with only deal with the population of observations Example: What is the probability of randomly selecting
If the distribution of a random variable x is approximately normal, then
Confidence Intervals for the Mean (σ unknown) In many real life situations, the standard deviation is unknown. In order to construct a confidence interval for a random variable that is normally distributed
Since his score is positive, he s above average. Since his score is not close to zero, his score is unusual.
Chapter 06: The Standard Deviation as a Ruler and the Normal Model This is the worst chapter title ever! This chapter is about the most important random variable distribution of them all the normal distribution.
Uniform Probability Distribution. Continuous Random Variables &
Continuous Random Variables & What is a Random Variable? It is a quantity whose values are real numbers and are determined by the number of desired outcomes of an experiment. Is there any special Random
Confidence Intervals and Sample Size
Confidence Intervals and Sample Size Chapter 6 shows us how we can use the Central Limit Theorem (CLT) to 1. estimate a population parameter (such as the mean or proportion) using a sample, and. determine
Math 14, Homework 7.1 p. 379 # 7, 9, 18, 20, 21, 23, 25, 26 Name
7.1 p. 379 # 7, 9, 18, 0, 1, 3, 5, 6 Name 7. Find each. (a) z α Step 1 Step Shade the desired percent under the mean statistics calculator to 99% confidence interval 3 1 0 1 3 µ 3σ µ σ µ σ µ µ+σ µ+σ µ+3σ
Chapter 7. Confidence Intervals and Sample Sizes. Definition. Definition. Definition. Definition. Confidence Interval : CI. Point Estimate.
Chapter 7 Confidence Intervals and Sample Sizes 7. Estimating a Proportion p 7.3 Estimating a Mean µ (σ known) 7.4 Estimating a Mean µ (σ unknown) 7.5 Estimating a Standard Deviation σ In a recent poll,
Elementary Statistics
Chapter 7 Estimation Goal: To become familiar with how to use Excel 2010 for Estimation of Means. There is one Stat Tool in Excel that is used with estimation of means, T.INV.2T. Open Excel and click on
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Chapter 6 Exam A Name The given values are discrete. Use the continuity correction and describe the region of the normal distribution that corresponds to the indicated probability. 1) The probability of
Chapter 6 Confidence Intervals Section 6-1 Confidence Intervals for the Mean (Large Samples) Estimating Population Parameters
Chapter 6 Confidence Intervals Section 6-1 Confidence Intervals for the Mean (Large Samples) Estimating Population Parameters VOCABULARY: Point Estimate a value for a parameter. The most point estimate
The Central Limit Theorem (Solutions) COR1-GB.1305 Statistics and Data Analysis
The Central Limit Theorem (Solutions) COR1-GB1305 Statistics and Data Analysis 1 You draw a random sample of size n = 64 from a population with mean µ = 50 and standard deviation σ = 16 From this, you
PREMIUM VERSION PREVIEW
FINANCIAL MATHS PREMIUM VERSION PREVIEW WWW.MATHSPOINTS.IE/SIGN-UP/ 205 LCHL Paper Question 6 (a) (i) Donagh is arranging a loan and is examining two different repayment options. Bank A will charge him
5.1 Personal Probability
5. Probability Value Page 1 5.1 Personal Probability Although we think probability is something that is confined to math class, in the form of personal probability it is something we use to make decisions
A) The first quartile B) The Median C) The third quartile D) None of the previous. 2. [3] If P (A) =.8, P (B) =.7, and P (A B) =.
![A) The first quartile B) The Median C) The third quartile D) None of the previous. 2. [3] If P (A) =.8, P (B) =.7, and P (A B) =.](/thumbs/95/124846909.jpg "A) The first quartile B) The Median C) The third quartile D) None of the previous. 2. [3] If P (A) =.8, P (B) =.7, and P (A B) =.")
Review for stat2507 Final (December 2008) Part I: Multiple Choice questions (on 39%): Please circle only one choice. 1. [3] Which one of the following summary measures is affected most by outliers A) The
MA131 Lecture 9.1. = µ = 25 and σ X P ( 90 < X < 100 ) = = /// σ X
The Central Limit Theorem (CLT): As the sample size n increases, the shape of the distribution of the sample means taken with replacement from the population with mean µ and standard deviation σ will approach
Section 7.5 The Normal Distribution. Section 7.6 Application of the Normal Distribution
Section 7.6 Application of the Normal Distribution A random variable that may take on infinitely many values is called a continuous random variable. A continuous probability distribution is defined by
Chapter 7. Inferences about Population Variances
Chapter 7. Inferences about Population Variances Introduction () The variability of a population s values is as important as the population mean. Hypothetical distribution of E. coli concentrations from
The Central Limit Theorem
Section 6-5 The Central Limit Theorem I. Sampling Distribution of Sample Mean ( ) Eample 1: Population Distribution Table 2 4 6 8 P() 1/4 1/4 1/4 1/4 μ (a) Find the population mean and population standard
Chapter 11: Inference for Distributions Inference for Means of a Population 11.2 Comparing Two Means
Chapter 11: Inference for Distributions 11.1 Inference for Means of a Population 11.2 Comparing Two Means 1 Population Standard Deviation In the previous chapter, we computed confidence intervals and performed
Hypothesis Tests: One Sample Mean Cal State Northridge Ψ320 Andrew Ainsworth PhD
Hypothesis Tests: One Sample Mean Cal State Northridge Ψ320 Andrew Ainsworth PhD MAJOR POINTS Sampling distribution of the mean revisited Testing hypotheses: sigma known An example Testing hypotheses:
Abraham Baldwin Agricultural College Math 2000 Practice Test 3
Abraham Baldwin Agricultural College Math 000 Practice Test 3 To get credit you must show your work. If your answer involves using a formula, make sure you write down the formula! Also write down any expression
6. THE BINOMIAL DISTRIBUTION
6. THE BINOMIAL DISTRIBUTION Eg: For 1000 borrowers in the lowest risk category (FICO score between 800 and 850), what is the probability that at least 250 of them will default on their loan (thereby rendering
Fall 2011 Exam Score: /75. Exam 3
Math 12 Fall 2011 Name Exam Score: /75 Total Class Percent to Date Exam 3 For problems 1-10, circle the letter next to the response that best answers the question or completes the sentence. You do not
As you draw random samples of size n, as n increases, the sample means tend to be normally distributed.
The Central Limit Theorem The central limit theorem (clt for short) is one of the most powerful and useful ideas in all of statistics. The clt says that if we collect samples of size n with a "large enough
Confidence Intervals: Review
University of Utah February 28, 2018 1 2 Law of Large Numbers Draw your samples from any population with finite mean µ. Then LLN says Law of Large Numbers Draw your samples from any population with finite
Study Ch. 7.3, # 63 71
GOALS: 1. Understand the distribution of the sample mean. 2. Understand that using the distribution of the sample mean with sufficiently large sample sizes enables us to use parametric statistics for distributions
Using the Central Limit Theorem It is important for you to understand when to use the CLT. If you are being asked to find the probability of the
Using the Central Limit Theorem It is important for you to understand when to use the CLT. If you are being asked to find the probability of the mean, use the CLT for the mean. If you are being asked to
AP 9.2 Notes WEB.notebook February 04, Bellwork
Bellwork A contract between a manufacturer and a consumer of light bulbs specifies that with an SRS of 100 bulbs. (a) Describe what a Type I error would be in this context. (b) Describe what a Type II
WebAssign Math 3680 Homework 5 Devore Fall 2013 (Homework)
WebAssign Math 3680 Homework 5 Devore Fall 2013 (Homework) Current Score : 135.45 / 129 Due : Friday, October 11 2013 11:59 PM CDT Mirka Martinez Applied Statistics, Math 3680-Fall 2013, section 2, Fall
. 13. The maximum error (margin of error) of the estimate for μ (based on known σ) is:
Statistics Sample Exam 3 Solution Chapters 6 & 7: Normal Probability Distributions & Estimates 1. What percent of normally distributed data value lie within 2 standard deviations to either side of the
Chapter 6: The Normal Distribution
Chapter 6: The Normal Distribution Diana Pell Section 6.1: Normal Distributions Note: Recall that a continuous variable can assume all values between any two given values of the variables. Many continuous
Chapter Four: Introduction To Inference 1/50
Chapter Four: Introduction To Inference 1/50 4.1 Introduction 2/50 4.1 Introduction In this chapter you will learn the rationale underlying inference. You will also learn to apply certain inferential techniques.
ECON 214 Elements of Statistics for Economists 2016/2017
ECON 214 Elements of Statistics for Economists 2016/2017 Topic The Normal Distribution Lecturer: Dr. Bernardin Senadza, Dept. of Economics bsenadza@ug.edu.gh College of Education School of Continuing and
Midterm Exam III Review
Midterm Exam III Review Dr. Joseph Brennan Math 148, BU Dr. Joseph Brennan (Math 148, BU) Midterm Exam III Review 1 / 25 Permutations and Combinations ORDER In order to count the number of possible ways
* Point estimate for P is: x n
Estimation and Confidence Interval Estimation and Confidence Interval: Single Mean: To find the confidence intervals for a single mean: 1- X ± ( Z 1 σ n σ known S - X ± (t 1,n 1 n σ unknown Estimation
d) Find the standard deviation of the random variable X.
Q 1: The number of students using Math lab per day is found in the distribution below. x 6 8 10 12 14 P(x) 0.15 0.3 0.35 0.1 0.1 a) Find the mean for this probability distribution. b) Find the variance
Lecture 39 Section 11.5
on Lecture 39 Section 11.5 Hampden-Sydney College Mon, Nov 10, 2008 Outline 1 on 2 3 on 4 on Exercise 11.27, page 715. A researcher was interested in comparing body weights for two strains of laboratory
ECO220Y Continuous Probability Distributions: Normal Readings: Chapter 9, section 9.10
ECO220Y Continuous Probability Distributions: Normal Readings: Chapter 9, section 9.10 Fall 2011 Lecture 8 Part 2 (Fall 2011) Probability Distributions Lecture 8 Part 2 1 / 23 Normal Density Function f
Confidence Intervals for the Mean. When σ is known
Confidence Intervals for the Mean When σ is known Objective Find the confidence interval for the mean when s is known. Intro Suppose a college president wishes to estimate the average age of students attending
Module 4: Probability
Module 4: Probability 1 / 22 Probability concepts in statistical inference Probability is a way of quantifying uncertainty associated with random events and is the basis for statistical inference. Inference
AP * Statistics Review
AP * Statistics Review Normal Models and Sampling Distributions Teacher Packet AP* is a trademark of the College Entrance Examination Board. The College Entrance Examination Board was not involved in the
Expected Value of a Random Variable
Knowledge Article: Probability and Statistics Expected Value of a Random Variable Expected Value of a Discrete Random Variable You're familiar with a simple mean, or average, of a set. The mean value of
Lecture 37 Sections 11.1, 11.2, Mon, Mar 31, Hampden-Sydney College. Independent Samples: Comparing Means. Robb T. Koether.
: : Lecture 37 Sections 11.1, 11.2, 11.4 Hampden-Sydney College Mon, Mar 31, 2008 Outline : 1 2 3 4 5 : When two samples are taken from two different populations, they may be taken independently or not
Distribution of the Sample Mean
Distribution of the Sample Mean MATH 130, Elements of Statistics I J. Robert Buchanan Department of Mathematics Fall 2018 Experiment (1 of 3) Suppose we have the following population : 4 8 1 2 3 4 9 1
The Central Limit Theorem for Sample Means (Averages)
The Central Limit Theorem for Sample Means (Averages) By: OpenStaxCollege Suppose X is a random variable with a distribution that may be known or unknown (it can be any distribution). Using a subscript
Normal distribution. We say that a random variable X follows the normal distribution if the probability density function of X is given by
Normal distribution The normal distribution is the most important distribution. It describes well the distribution of random variables that arise in practice, such as the heights or weights of people,
Estimation and Confidence Intervals
Estimation and Confidence Intervals Chapter 9-1/2 McGraw-Hill/Irwin Copyright 2011 by the McGraw-Hill Companies, Inc. All rights reserved. LEARNING OBJECTIVES LO1. Define a point estimate. LO2. Define
No, because np = 100(0.02) = 2. The value of np must be greater than or equal to 5 to use the normal approximation.
1) If n 100 and p 0.02 in a binomial experiment, does this satisfy the rule for a normal approximation? Why or why not? No, because np 100(0.02) 2. The value of np must be greater than or equal to 5 to
1. Variability in estimates and CLT
Unit3: Foundationsforinference 1. Variability in estimates and CLT Sta 101 - Fall 2015 Duke University, Department of Statistical Science Dr. Çetinkaya-Rundel Slides posted at http://bit.ly/sta101_f15
CHAPTER 6 Random Variables
CHAPTER 6 Random Variables 6.1 Discrete and Continuous Random Variables The Practice of Statistics, 5th Edition Starnes, Tabor, Yates, Moore Bedford Freeman Worth Publishers Discrete and Continuous Random
2.) What is the set of outcomes that describes the event that at least one of the items selected is defective? {AD, DA, DD}
Math 361 Practice Exam 2 (Use this information for questions 1 3) At the end of a production run manufacturing rubber gaskets, items are sampled at random and inspected to determine if the item is Acceptable
VARIABILITY: Range Variance Standard Deviation
VARIABILITY: Range Variance Standard Deviation Measures of Variability Describe the extent to which scores in a distribution differ from each other. Distance Between the Locations of Scores in Three Distributions
CH 5 Normal Probability Distributions Properties of the Normal Distribution
Properties of the Normal Distribution Example A friend that is always late. Let X represent the amount of minutes that pass from the moment you are suppose to meet your friend until the moment your friend
Chapter 7 presents the beginning of inferential statistics. The two major activities of inferential statistics are
Chapter 7 presents the beginning of inferential statistics. Concept: Inferential Statistics The two major activities of inferential statistics are 1 to use sample data to estimate values of population
Lecture 6: Chapter 6
Lecture 6: Chapter 6 C C Moxley UAB Mathematics 3 October 16 6.1 Continuous Probability Distributions Last week, we discussed the binomial probability distribution, which was discrete. 6.1 Continuous Probability
Normal Sampling and Modelling
8.3 Normal Sampling and Modelling Many statistical studies take sample data from an underlying normal population. As you saw in the investigation on page 422, the distribution of the sample data reflects
Determine whether the given procedure results in a binomial distribution. If not, state the reason why.
Math 5.3 Binomial Probability Distributions Name 1) Binomial Distrbution: Determine whether the given procedure results in a binomial distribution. If not, state the reason why. 2) Rolling a single die