...hypothesis test of a proportion, when the following conditions are met: * The sampling method is simple random sampling. * Each sample point can result in just two possible outcomes. We call one of these outcomes a success and the other, a failure. * The sample includes at least 10 successes and 10 failures. (Some texts say that 5 successes and 5 failures are enough.) * The population size is at least 10 times as big as the sample size. This approach consists of four steps: (1) state the hypotheses, (2) formulate an analysis plan, (3) analyze sample data, and (4) interpret results. State the Hypotheses Every hypothesis test requires the analyst to state a null hypothesis and an alternative hypothesis. The hypotheses are stated in such a way that they are mutually exclusive. That is, if one is true, the other must be false; and vice versa. Formulate an Analysis Plan The analysis plan describes how to use sample data to accept or reject the null hypothesis. It should specify the following elements. * Significance level. Often, researchers choose significance levels equal to 0.01, 0.05, or 0.10; but any value between 0 and 1 can be used. * Test method. Use the one-sample z-test to determine whether the hypothesized population proportion differs significantly from the observed sample proportion. Analyze Sample Data Using sample data, find the test statistic and its associated P-Value. * Standard deviation. Compute the standard deviation (σ) of the sampling...
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...MEASUREMENT IN MARKETING RESEARCH Chapter 8 Brand loyalty – can be defined as the last brand acquired, or can be defined as the person’s most preferred brand. 3 Basic Question-Response Formats 1. Open-ended question – presents no response options to the respondent. Rather, the respondent is instructed to responds in his/her own word. 2 Types of Open-ended Question a. Unprobed format – seeks no additional information from the respondent. b. Probed format – the researcher wants a comment or statement from the respondent, or simply wants the respondent to indicate a name of a brand or a store. Response probe – instructing the interviewer to ask for additional information. Pros and Cons of Alternative Response Formats Response Format Examples: Pros (+) and Cons (-) Unprobed open-ended question “What was your reaction to the Sony CD player advertisement you saw on TV last?” + Allows respondent to use his/her own words. - Difficult to code and interpret. - Respondents may not give complete answer. Probe open-ended question “Did you have any other thoughts or reactions to the advertisement?” + Elicits complete answers. - Difficult to code and interpret Dichotomous closed-ended question “Do...
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...Lab 61: Confidence Intervals on Proportions In this lab you're going to use the simulation at http://statweb.calpoly.edu/chance/applets/Reeses/ReesesPieces.html to take virtual samples of Reese's pieces (sorry no real chocolate) Open up the simulator and set π = .3. This is really p, but on my screen it comes out as π, set Sample Size, n = 30 . This means we are setting the proportion of orange Reese's pieces as .3 When you click on "Select Sample" the computer will pull 30 Reese's pieces, and put them in bins as orange, yellow, and dark brown and it will tell you p for orange. What you're going to do is repeatedly sample, using the simulator, and after each sample calculate confidence intervals for confidence levels of 60%, 80% and 99%. You'll (ok, the computer, will do a total of 20 samples). Then we'll see how many confidence intervals at each level contain the actual population proportion. Some formulas and items you will need: CI = p± z*σ, where σ=p(1-p)n Remember that p can be given in the problem or a calculated p. Use the simulator to find p , then calculate the rest to fill in the table. For this lab, σ= .084 If you calculate the # orange, you can use stat-tests-1propZint on your calculator Sample # | p orange | # orange = 30p | 60% Confidence Interval z*=.84 | 80% Confidence Interval z*=1.28 | 99% Confidence Interval z*=2.58 | | | | Min | Max | Min | Max | Min | Max | 1 | | | | | | | | | 2 | | | | | | | ...
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...mean weight of a random sample of n apples of a particular kind is to be used to estimate the mean weight of all such apples. If σ= 1.5 oz and the probability is to be 0.95 that the estimate is in error by at most 0.5 oz, then n must be equal to: | | 24 | | | 25 | | | 34 | | | 35 | A sample size of 14 is used to find a confidence interval. The number of degrees of freedom needed is 13 A small sample confidence interval formula for a population mean differs from the corresponding large sample formula in that the small sample formula contains a ____________ The sample standard deviation S may be used in place of σ in the large sample confidence interval for N provided that n is at least _____________ . Consider a large population with a mean of 150 and a standard deviation of 27. A random sample of size 36 is taken from this population. The standard error of the sampling distribution of sample mean is equal to: | | 4.17 | | | 5.20 | | | 4.50 | | | 5.56 | Which of the following is NOT true for the t distribution ? | | Its shape depends on the number of degrees of freedom | | | Its mean depends on the number of degrees of freedom | | | it is symmetrical | | | It is bell shaped | The endpoints of a confidence interval are called the confidence limits A single number which is used as an estimate of an unknown parameter is called a _________estimate Point To obtain a large sample confidence interval for...
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...hypothesis about a parameter in a population, using data measured in a sample. In this method, we test some hypothesis by determining the likelihood that a sample statistic could have been selected, if the hypothesis regarding the population parameter were true. Hypothesis testing or significance testing is a method for testing a claim or hypothesis about a parameter in a population, using data measured in a sample. In this method, we test some hypothesis by determining the likelihood that a sample statistic could have been selected, if the hypothesis regarding the population parameter were true. The method of hypothesis testing can be summarized in four steps. 1. To begin, we identify a hypothesis or claim that we feel should be tested. For example, we might want to test the claim that the mean number of hours that children in the United States watch TV is 3 hours. 2. We select a criterion upon which we decide that the claim being tested is true or not. For example, the claim is that children watch 3 hours of TV per week. Most samples we select should have a mean close to or equal to 3 hours if the claim we are testing is true. So at what point do we decide that the discrepancy between the sample mean and 3 is so big that the claim we are testing is likely not true? We answer this question in this step of hypothesis testing. 3. Select a random sample from the population and measure the sample mean. For example, we could select 20 children and measure the...
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...the quiz questions as shown and the tutorial questions. Week 1 (Descriptive Statistics) Concepts • Population vs. sample • Parameter vs. statistic and use of correct symbols • Types of data (nominal, ordinal, interval) • Displaying data: Nominal: Bar and pie charts Interval: Stem and leaf displays, histograms, ogives, box and whisker plots • Measures of central location (mean, median, mode) • Measures of dispersion/spread (standard deviation, variance) Do 1.7.3 Quiz Can attempt all 1.7.4 Topic Questions Week 2 (Sampling, Probability, Normal Distribution) Sampling Concepts • Types of sampling plans (simple random sampling, stratified random sampling, cluster sampling Probability Concepts • From a table of joint probabilities answer such questions as: P(A and B) P(A or B) P(A)|B Determine if events A and B are mutually exclusive or not Determine if events A and B are independent or not The Normal Distribution Question gives a population mean and population standard deviation for a normally distributed variable and asks such questions as: • P(X > or ≥ a value of the variable) • P(X < or ≤ a value of the variable) • P(a value of the variable ≤ X ≤ a value of the variable) The formula for the normal distribution is: [pic] Do 2.7.2 Quiz Attempt Do all 2.7.3 Topic Questions...
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...An Introduction to
Statistics
Keone Hon
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...contingency tables. In 2000 the Vermont State legislature approved a bill authorizing civil unions. The vote can be broken down by gender to produce the following table, with the expected frequencies given in parentheses. The expected frequencies are computed as Ri × Cj /N, where Ri and Cj represent row and column marginal totals and N is the grand total. Vote Women Men Total Yes 35 (28.83) 60 (66.17) 95 No 9 (15.17) 41 (34.83) 50 Total 44 101 145 The standard Pearson chi-square statistic is defined as χ2 = (Oij − Eij )2 (35 − 28.83)2 (41 − 34.83)2 = + ··· + = 5.50 Eij 28.83 34.83 where i and j index the rows and columns of the table. (For the goodness-of-fit test we simply drop the subscript j.) The resulting test statistic from the formula on the left is approximately distributed as χ2 on (r − 1)(c − 1) degrees of freedom. 1 The probability of χ2 > 5.50 on 1 df = .019, so we can reject the null hypothesis that voting behavior is independent of gender. (Pearson originally misidentified the degrees of freedom,...
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...to establish its truth. 3. Test Statistic - A sample statistic (often a formula) that is used to decide whether to reject H0 . 4. Rejection Region- It consists of all values of the test statistic for which H0 is rejected. This rejection region is selected in such a way that the probability of rejecting true H0 is equal to α (a small number usually 0.05). The value of α is referred to as the level of significance of the test. 5. Assumptions - Statements about the population(s) being sampled. 6. Calculation of the test statistic and conclusion- Reject H0 if the calculated value of the test statistic falls in the rejection region. Otherwise, do not reject H0 . 7. P-value or significance probability is defined as proportion of samples that would be unfavourable to H0 (assuming H0 is true) if the observed sample is considered unfavourable to H0 . If the p-value is smaller than α, then reject H0 . Remark: 1. If you fix α = 0.05 for your test, then you are allowed to reject true null hypothesis 5% of the time in repeated application of your test rule. 2. If the p-value of a test is 0.20 (say) and you reject H0 then, under your test rule, at least 20% of the time you would reject true null hypothesis. 1. Large sample (n > 30) test for H0 : µ = µ0 (known). Z= x − µ0 ¯ σ √ n Example. A study reported in the Journal of Occupational and Organizational Psychology investigated the relationship of employment status to mental health. Each of a sample of 49 unemployed men was given a mental health...
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...Chapter 20 Statistical Quality Control GOALS When you have completed this chapter, you will be able to: • Discuss the role of quality control in production and service operations • Define and understand the terms chance cause, assignable cause, in control, out of control, attribute, and variable • Construct and interpret a Pareto chart • Construct and interpret a fishbone diagram • Construct and interpret a mean and range chart • Construct and interpret a percent defective and a c-bar chart • • Discuss acceptance sampling Construct an operating characteristic curve for various sampling plans. W ter A. Shewhart (1891–1967) al W 1498 1548 1598 1648 1698 1748 1898 1948 2000 ith the advent of industrial revolution in the 19th century, mass production replaced manufacturing in small shops by skilled craftsman and artisans. While in the small shops the individual worker was completely responsible for the quality of the work, this was no longer true in mass production where each individual’s contribution to the finished product constituted only an insignificant part in the total process. The quality control by the large companies was achieved with the help of quality inspectors responsible for checking a 100 percent inspection of all the important characteristics. Dr. Walter A. Shewhart, called the father of quality control analysis, developed the concepts of statistical quality control. For the purpose of controlling quality, Shewhart developed charting techniques and...
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...inference is to draw conclusion about the population parameters based on samples studies that is quite small in comparison to the size of the population. In order that conclusion of sampling theory and statistical inference valid, samples must be chosen so as to the representation of a population. For example, Television executives want to know the proportion of television viewers who watch that network’s program. Particularly determining the proportions that are watching certain programs is impractical and prohibitively expensive. One possible alternative method can be providing approximation by observing what a sample of 1,000 television viewer’s watch. Thus they estimate the population proportion by calculating the sample proportion. Similarly, the field of quality control illustrates yet another reason for sampling. In order to ensure that a production process is operating properly, the operations manager needs to know the proportion of defective units that are being produced. If the quality-control technician must destroy the unit in order to determine whether or not it is defective, there is no alternative to sampling: a complete inspection of the population would destroy the entire output of the production process. We know that the sample proportion of television viewers or of defective items is probably not exactly equal to the population proportion we want it to estimate. Nonetheless, the sample statistic can come quite close to the parameter it is designed to estimate...
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...This paper is not to be removed from the Examination Halls UNIVERSITY OF LONDON ST104A ZB (279 004A) BSc degrees and Diplomas for Graduates in Economics, Management, Finance and the Social Sciences, the Diplomas in Economics and Social Sciences and Access Route Statistics 1 (half unit) Friday, 4 [Month] 2012 : ##.##Xm to ##.##Xm [Day], ## May 2012 : 10.00am to 12.00pm Candidates should answer THREE of the following FOUR questions: QUESTION 1 of Section A (50 marks) and TWO questions from Section B (25 marks each). Candidates are strongly advised to divide their time accordingly. A list of formulae and extracts from statistical tables are given after the final question on this paper. Graph paper is provided at the end of this question paper. If used, it must be detached and fastened securely inside the answer book. A calculator may be used when answering questions on this paper and it must comply in all respects with the specification given with your Admission Notice. The make and type of machine must be clearly stated on the front cover of the answer book. © University of London 2012 UL12/0218 D01 PLEASE TURN OVER Page 1 of 21 SECTION A Answer all parts of Question 1 (50 marks in total). 1. (a) The following data represent different types of variables. Classify each one of them as measurable (continuous) or categorical. If a variable is categorical, further classify it as nominal or ordinal. Justify your answer. (Note that no marks...
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...misstatement exists in a materially misstated pop based on taking a sample that includes no misstatement;-2-In assessing sampling risk, the risk of incorrect rejection n the risk of assessing control risk too high-efficiency of the audit;-3-Which of the ff statistical sampling techni is least desirable 4 use by the auditors-block selection;-4-The auditors’ primary objective in selecting a sample of items 4m an audit pop is 2 obtain-a rep sample;-5-Discovery sampling is part eff when-the auditors r looking 4 critical deviations that r not expected 2 b frequent in no;-6-The auditors r using unstratified mean-per-unit sampling to audit AR as they did in the prior year. Which of the ff changes in chara or speci would result in a larger required sample size this yr than that required in the prior yr-larger variance in the $ value of accts.;-7-Which of the ff sampling tech is typically used 4 tests of contr-attribute sampling;-8-Which of the ff is accurate regarding tolerable misstatement-tolerable misstat is directly related to materiality;-9-In which of the ff circum is 8 least likely that tests of contr will b perform-the expected deviation rate exceeds the tolerable deviation rate.;-10-An auditor needs to estimate the average highway weight of tractor-trailer trucks using a state’s highway system. Which estimation mth would b most appropria?-mean-per-unit;-11-The auditors have sampled 50 a/cs 4m a pop of 1,000 AR. The sample items have a mean bk value of $200 n a mean audited value of...
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...1.1) Definition of 'Mode' A statistical term that refers to the most frequently occurring number found in a set of numbers. The mode is found by collecting and organizing the data in order to count the frequency of each result. The result with the highest occurrences is the mode of the set. Other related terms include the mean, or the average of a set; and the median, or the middle value in a set. Investopedia Says Investopedia explains 'Mode' For example, in the following list of numbers, 16 is the mode since it appears more times than any other number in the set: 3, 3, 6, 9, 16, 16, 16, 27, 27, 37, 48 A set of numbers can have more than one mode (this is known as bimodal) if there are multiple numbers that occur with equal frequency, and more times than the others in the set. 3, 3, 3, 9, 16, 16, 16, 27, 37, 48 In this example, both the number 3 and the number 16 are modes. If no number in a set of numbers occurs more than once, that set has no mode: 3, 6, 9, 16, 27, 37, 48 Fashion is generally an answer to the question, "What is the mode?" e.g. "What's the most popular sneaker?" or "What shirt style would most people want to wear?" 1.2) Definition of 'Median' The middle number in a sorted list of numbers. To determine the median value in a sequence of numbers, the numbers must first be arranged in value order from lowest to highest. If there is an odd amount of numbers, the median value is the number that is in the middle, with the same amount of numbers below and above....
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...Task 1 explores a little bit into confidence interval and how the width can be affected by the sample size. Students are expected to acquire the mean (average) of a normal population with a population variance of 1 and a confidence interval of 95%. In this task, we will see the relationship between sample size and width in detail by plotting graphs. We will also explore on the accuracy of an interval and how it can be increased or decreased by varying the value of the width. Variance, σ^2 = 1 Standard deviation, σ = 1 Confidence Interval: 95% From the formula to obtain width, The sample sizes of n = 25, 49, 64, 100 and 400 are substituted into the formula above to obtain the other values of width. The values are tabulated in a table below. hence the graph does not pass through the origin. It can be stated that as the sample size increases, the width will decrease. The higher the sample size,...
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