...Chapter 10 Statistical Inference About Means and Proportions with Two Populations Learning Objectives 1. Be able to develop interval estimates and conduct hypothesis tests about the difference between two population means whenandare known. 2. Know the properties of the sampling distribution of . 3. Be able to use the t distribution to conduct statistical inferences about the difference between two population means whenandare unknown. 4. Learn how to analyze the difference between two population means when the samples are independent and when the samples are matched. 5. Be able to develop interval estimates and conduct hypothesis tests about the difference between two population proportions. 6. Know the properties of the sampling distribution of . ...
Words: 3502 - Pages: 15
...20 | 5.0 Points | When comparing two population means with an unknown standard deviation you use a t test and you use N-2 degrees of freedom. A. True | B. False | | Reset Selection Question 2 of 20 | 5.0 Points | Pretend you want to determine whether the mean weekly sales of soup are the same when the soup is the featured item and when it is a normal item on the menu. When it is the featured item the sample mean is 66 and the population standard deviation is 3 with a sample size of 23. When it is a normal item the sample mean is 53 with a population standard deviation of 4 and a sample size of 7. Given this information we could use a t test for two independent means. A. True | B. False | | Reset Selection Question 3 of 20 | 5.0 Points | The alternative hypothesis can be proven if the alternative hypothesis is rejected. A. True | B. False | | Reset Selection Question 4 of 20 | 5.0 Points | You want to determine if your widgets from machine 1 are the same as machine 2. Machine 1 has a sample mean of 50 and a population standard deviation 5 and a sample size of 100. Machine 2 has a sample mean of 52 and a population standard deviation of 6 with a sample size of 36. With an alpha of .10 can we claim that there is a difference between the output of the two machines. Which of the following statements are true? A. We will reject the null hypothesis and prove there is a difference between the 2 populations | | B. We will not reject the...
Words: 1999 - Pages: 8
...Chapter 10: Comparing Two Groups Bivariate Analysis: Methods for comparing two groups are special cases of bivariate statistical methods – Two variables exist: Response variable – outcome variable on which comparisons are made Explanatory variable – binary variable that specifies the groups Statistical methods analyze how the outcome on the response variable depends on or is explained by the value of the explanatory variable Independent Samples: Most comparisons of groups use independent samples from the groups, The observations in one sample are independent of those in the other sample Example: Randomized experiments that randomly allocate subjects to two treatments Example: An observational study that separates subjects into groups according to their value for an explanatory variable Dependent samples: Dependent samples result when the data are matched pairs – each subject in one sample is matched with a subject in the other sample Example: set of married couples, the men being in one sample and the women in the other. Example: Each subject is observed at two times, so the two samples have the same subject Categorical response variable: For a categorical response variable - Inferences compare groups in terms of their population proportions in a particular category - We can compare the groups by the difference in their population proportions: (p1 – p2) Example: Experiment: Subjects were 22,071 male physicians Every other day for five years, study participants...
Words: 6772 - Pages: 28
...assembly line. For each inspector, the number of components that can be checked in a shift can be represented by a random variable with mean 120 and standard deviation 16. Let X represent the number of components checked by an inspector in a shift. Then the total number checked is 5X, which has mean 600 and standard deviation 80. What is wrong with this argument? Assuming that inspectors' performances are independent of one another, find the mean and standard deviation of the total number of components checked in a shift. The calculation for the mean is OK, but the calculation of the standard deviation is not. Remember that a standard deviation is a square root, and that: |[pic][pic] |[pic] | We need to do our algebra on the variance, not the standard deviation. Therefore: |[pic] |[pic] | | |[pic] | | |[pic] | | |[pic] | 2. It is estimated that in normal highway driving, the number of miles that can be covered by automobiles of a particular model on 1 gallon of gasoline can be represented by a random variable with mean 28 and standard deviation 2.4. Sixteen of these cars, each with 1 gallon of gasoline, are driven independently under highway conditions. Find the mean and standard deviation of the average number of miles that will be achieved by these cars. |[pic] |[pic] ...
Words: 3946 - Pages: 16
...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 mean time (in...
Words: 13735 - Pages: 55
...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 distribution. σ = sqrt[ P * ( 1 - P ) / n ] where P is the hypothesized value of population proportion in the null hypothesis...
Words: 3650 - Pages: 15
...Airline carrier flight times vary slightly from carrier to carrier and even when the airport origins and destinations are the same. Is there a mean difference between San Francisco and Dallas Fort Worth Texas airport in the USA when comparing the flight time between Sky West (OO) and Virgin airlines (VX)? To answer this question, two randomly sampled groups were used to collect data for a two-sample test to conduct a hypothesis which would provide an answer about the mean. I will compare the means of Sky West and Virgin America Airline flight time between San Francisco and Dallas Fort Worth and compare them. I expect the test to reveal the airline with the fastest flight times between the two cities. Population The population that I intend to study is Sky West and Virgin Airlines. I will compare the two airlines and determine whether it’s faster to fly from San Francisco to Dallas Fort Worth using the air time in this T-test. Variables The two sample T test will use the air times variable for either Sky west Airlines or Virgin Airline to fly from San Francisco, California (SFO) airport to Dallas Fort Worth, Texas (DFW) airport. Data Collection Data for this T test was collected from a file generated on the Research and Innovative Technology Administration Bureau of Transportation Statistics website, between two airlines, Sky West Airline and Virgin America Airlines flight between (SFO) San Francisco, CA International Airport and (DFW), Dallas Forth Worth...
Words: 1155 - Pages: 5
...variable is a person's rating of someone else's attractiveness on a 4 point scale. Categorical Variables Categorical variables represent types of data which may be divided into groups. Examples of categorical variables are race, sex, age group, and educational level. While the latter two variables may also be considered in a numerical manner by using exact values for age and highest grade completed, it is often more informative to categorize such variables into a relatively small number of groups. Analysis of categorical data generally involves the use of data tables. A two samples or measures presents categorical data by counting the number of observations that fall into each group for two variables, one divided into rows and the other divided into columns. For example, a survey to identify their hair and eye color. Continuous Variables Much of the statistical analysis in medical research involves the analysis of continuous variables such as cardiac output, blood pressure, and heart rate which can assume an infinite range of values. As with discrete variables, the statistical analysis of continuous variables requires the application of specialized tests. These tests compare the means of two or more data sets to determine whether the data sets differ significantly from one another. There are four...
Words: 1276 - Pages: 6
...about a population parameter. In the same way, a point estimate of the mean overpayment is simply a good guess about what the average overpayment for the population is. Investigating all 1,000 claims and obtaining the overpayment amount for each would either be impractical, unfeasible or both. Thus, the auditor deems a sample size of 50 claims to be adequate and sufficiently representative of the entire population. The mean overpayment amount of this sample is then calculated in order to obtain a point estimate of the mean overpayment. The estimated mean is then extrapolated to the overpayment amount to the population of all 1,000 claims. http://pages.wustl.edu/montgomery/lecture-7 Point Estimate vs. Interval Estimate To estimate population parameters, statisticians use sample statistics. For example, we use sample means to estimate population means and we use sample proportions to estimate population proportions. An estimate of a population parameter can be expressed in one of two ways: * Point estimate. A point estimate of a population parameter is a single value of a statistic. For example, the sample mean x is a point estimate of the population mean μ. In the same way, a sample proportion p is a point estimate of the population proportion P. * Interval estimate. An interval estimate is defined by two numbers, and the population parameter is said to lie between those two numbers. For example, a < x < b represents an interval estimate of the population mean...
Words: 372 - Pages: 2
...describe a characteristic. (Numerical summary of the population) (Ex. The proportion of females randomly selected from the registered voters of a county or the average time it takes for the first 5 students to complete an exam) Parameter- A characteristic (numerical summary) of the entire population. A number computed from the entire population usually unobservable. Subject- the entities that we measure in a study. Population- the collection of all subjects under study. Proportion- of the observations that fall in a certain category is the frequency (count) of observations in that category divided by the total number of observations. Proportion and percentages (aka relative frequencies) Probability- the chance that an even will occur. Sample- a subset of the population. Random Samples are most valid. Design- Plan for how the data will be collected. Two events are dependent if the outcome or occurrence of the first affects the outcome or occurrence of the second so that the probability is changed. Sample Space- set of all possible outcomes for a random phenomenon Variables (COLUMNS)- characteristics or measurements on the subject. Event-subset of the sample space Description- summarizing the data that are obtained. Descriptive Statistics- methods for summarizing the data. Usually consists of bar graphs and numbers like averages and %’s. Inference- Making conclusions (predictions) about the population based on the data collected (Inferential Statistics)...
Words: 12484 - Pages: 50
...CHAPTER 9 Hypothesis Tests CONTENTS 9.4 POPULATION MEAN: σ UNKNOWN One-Tailed Test Two-Tailed Test Summary and Practical Advice 9.5 POPULATION PROPORTION Summary 9.6 HYPOTHESIS TESTING AND DECISION MAKING 9.7 CALCULATING THE PROBABILITY OF TYPE II ERRORS 9.8 DETERMINING THE SAMPLE SIZE FOR A HYPOTHESIS TEST ABOUT A POPULATION MEAN STATISTICS IN PRACTICE: JOHN MORRELL & COMPANY 9.1 DEVELOPING NULL AND ALTERNATIVE HYPOTHESES The Alternative Hypothesis as a Research Hypothesis The Null Hypothesis as an Assumption to Be Challenged Summary of Forms for Null and Alternative Hypotheses 9.2 TYPE I AND TYPE II ERRORS 9.3 POPULATION MEAN: σ KNOWN One-Tailed Test Two-Tailed Test Summary and Practical Advice Relationship Between Interval Estimation and Hypothesis Testing 349 Statistics in Practice STATISTICS in PRACTICE JOHN MORRELL & COMPANY* CINCINNATI, OHIO John Morrell & Company, which began in England in 1827, is considered the oldest continuously operating meat manufacturer in the United States. It is a wholly owned and independently managed subsidiary of Smithfield Foods, Smithfield, Virginia. John Morrell & Company offers an extensive product line of processed meats and fresh pork to consumers under 13 regional brands including John Morrell, E-Z-Cut, Tobin’s First Prize, Dinner Bell, Hunter, Kretschmar, Rath, Rodeo, Shenson, Farmers Hickory Brand, Iowa Quality, and Peyton’s. Each regional brand enjoys high brand recognition and loyalty among consumers. Market...
Words: 29095 - Pages: 117
...Department of Mathematics Created by Prof. Tan Le & Hang Lai STATISTICS – EXCEL 2010 INSTRUCTIONS Table of Contents 1. Open Microsoft Excel .............................................................................................................................................. 3 2. Bar Charts.................................................................................................................................................................... 4 3. Pie Charts..................................................................................................................................................................... 7 4. Line Charts ................................................................................................................................................................ 10 5. How to install the Data Analysis Package in Excel 2010? ..................................................................... 11 6. Descriptive Statistics ............................................................................................................................................ 12 7. Histograms ................................................................................................................................................................ 14 8. Making an Ogive ...........................................................................................................................
Words: 4093 - Pages: 17
...techniques has developed. Statistics is defined as the process of collecting a sample, organizing, analyzing and interpreting data. The numeric values which represent the characteristics analyzed in this process are also referred to as statistics. When information related to a particular group is desired, and it is impossible or impractical to obtain this information, a sample or subset of the group is obtained and the information of interest is determined for the subset. For instance someone is interested in the average annual income of all the students with majors in the College of Business Administration at Sam Houston State University, the only way this information could be obtained is if the annual income of every student in this population could be collected, recorded and analyzed without error. Since this would take considerable time and money, and since the...
Words: 5737 - Pages: 23
...Hypothesis Testing: One Population Hypothesis testing is used to make decisions about a population based on the analysis of sample statistics. There are always two hypothesis statements which are mutually exclusive and complementary statements concerning the value of the population parameter of interest: Null Hypothesis (H0): An assumption regarding a population parameter of interest. This hypothesis always includes the equal sign as the null hypothesis is assumed to be true unless sample data produces evidence to the contrary. Alternative Hypothesis (H1): The alternative option available when the null hypothesis is rejected. Three Types of Hypothesis Tests: Lower-Tail Test Upper-Tail Test Two-Tailed Test H0: ( ( (0 H0: ( ( (0 H0: ( = (0 H1: ( < (0 H1: ( > (0 H1: ( ( (0 Two-Sided Tests: A Two-Tailed Test (TTT) is designed to detect a change in the population parameter of interest from some hypothesized value to some different value. One-Sided Tests: A Lower-Tail Test (LTT) is designed to detect a change in the population parameter of interest from some hypothesized value to some lower value. An Upper-Tail Test (UTT) is designed to detect a change in the population parameter of interest from some hypothesized value to some greater value. Test Statistics and Action Limits: A test statistic is a sample statistic corresponding to the population parameter being tested. For example, when testing a hypothesis...
Words: 2155 - Pages: 9
...Type I and Type II errors. 4 Define the term test statistic and explain how it is used. 5 Distinguish between a one-tailed and a two-tailed hypothesis. 6 Conduct a test of hypothesis about a population mean. 7 Compute and interpret a p-value. 8 Conduct a test of hypothesis about a population proportion. 10-2 Define a hypothesis. Explain the five-step hypothesis-testing procedure. Hypothesis and Hypothesis Testing HYPOTHESIS A statement about the value of a population parameter developed for the purpose of testing. HYPOTHESIS TESTING A procedure based on sample evidence and probability theory to determine whether the hypothesis is a reasonable statement. 10-3 The Null and Alternate Hypotheses NULL HYPOTHESIS A statement about the value of a population parameter developed for the purpose of testing numerical evidence. ALTERNATE HYPOTHESIS A statement that is accepted if the sample data provide sufficient evidence that the null hypothesis is false. 10-4 Important Things to Remember about H0 and H1 H0: null hypothesis and H1: alternate hypothesis. H0 and H1 are mutually exclusive and collectively exhaustive. H0 is always presumed to be true. H1 has the burden of proof. A random sample (n) is used to “reject H0”. If we conclude “do not reject H0”, this does not necessarily mean that the null hypothesis is true, it only suggests that there is not sufficient evidence to reject H0; rejecting the null hypothesis...
Words: 2308 - Pages: 10
