BMAL 590 Quantitative Research Techniques and Statistics Exam 2023/2024, Exams of Statistics

Download BMAL 590 Quantitative Research Techniques and Statistics Exam 2023/2024 and more Exams Statistics in PDF only on Docsity! BMAL 590 Quantitative Research Techniques and Statistics Exam 2023/2024 | Actual Exam | Already Verified Answers When every possible sample with the same number of observations is equally likely to be chosen, the result is called a ----CORRECT ANSWER-----Simple random sample Which of the following types of samples is almost always biased? ----CORRECT ANSWER-----Self-selected samples Which of the following probabilities is equal to the significance level α? ----CORRECT ANSWER-----Probability of making a Type I error If we reject the null hypothesis, we conclude that ----CORRECT ANSWER-----There is enough statistical evidence to infer that the alternative hypothesis is true Statisticians can translate p-values into several descriptive terms. Suppose you typically reject H0 at level 0.05. Which of the following statements is correct? ----CORRECT ANSWER-----All of these choices are true The p-value of a test is the ----CORRECT ANSWER-----Smallest α at which the null hypothesis can be rejected The numerical quantity computed from the data that is used in deciding whether to reject H0 is the ----CORRECT ANSWER-----Test statistic For a given level of significance, if the sample size increases, the probability of a Type II error will ----CORRECT ANSWER-----Decrease The power of a test is measured by its capability of ----CORRECT ANSWER----- Rejecting a null hypothesis that is false If the probability of committing a Type I error for a given test is decreased, then for a fixed sample size n, the probability of committing a Type II error will ----CORRECT ANSWER-----Increase Which of the following is an example of a nonsampling error? ----CORRECT ANSWER-- ---All of these choices are true Which of the following situations lends itself to cluster samples? ----CORRECT ANSWER-----All of these choices are true Which of the following causes sampling error? ----CORRECT ANSWER-----Taking a random sample from a population instead of studying the entire population Which of the following describes selection bias? ----CORRECT ANSWER-----Some members of the target population are excluded from possible selection for the sample An approach of assigning probabilities which assumes that all outcomes of the experiment are equally likely is referred to as the ----CORRECT ANSWER-----Classical approach A company has developed a new computer sound card whose average lifetime is unknown. In order to estimate this average, 200 sound cards are randomly selected from a large production line and tested; their average lifetime is found to be 5 years. The five years represents a ----CORRECT ANSWER-----Statistics A summary measure that is computed from a population is called a ----CORRECT ANSWER-----Parameter Which of the following is a measure of the reliability of a statistical inference? ---- CORRECT ANSWER-----A significance level The process of using sample statistics to draw conclusions about population parameters is called ----CORRECT ANSWER-----Doing inferential statistics Which of the following represents a population, as opposed to a sample? ----CORRECT ANSWER-----All registered voters in the State of Michigan A study in under way to determine the average height of all 32,000 adult pine trees in a certain national forest. The heights of 500 randomly selected adult pine trees are measured and analyzed. The sample in this study is ----CORRECT ANSWER-----The 500 adult pine trees selected at random selected at random from this forest The significance level of a statistical inference measures ----CORRECT ANSWER----- The proportion of times a conclusion about a population will be wrong in the long run The confidence level of a statistical inference measures ----CORRECT ANSWER----- The proportion of times an estimation procedure will be correct in the long run A marketing research firm selects a random sample of adults and asks them a list of questions regarding their beverage preferences. What type of data collection is involved here? ----CORRECT ANSWER-----A survey Which of the following statements is true regarding the design of a good survey? ---- CORRECT ANSWER-----All of these choices are true A Type II error is defined as ----CORRECT ANSWER-----Not rejecting a false null hypothesis A robust estimator is one that is ----CORRECT ANSWER-----Not sensitive to moderate nonnormality For statistical inference about the mean of a single population when the population standard deviation is unknown, the degrees for freedom for the t-distribution equal n - 1 because we lose one degree of freedom by using the ----CORRECT ANSWER----- Sample mean as an estimate of the population mean The degrees of freedom for the test statistic for μ when σ is unknown is ----CORRECT ANSWER-----n - 1 The statistic (n - 1)s2 / σ2 has a chi-squared distribution with n - 1 degrees of freedom if ----CORRECT ANSWER-----the population is normally distributed with variance equal to σ2 Which of the following is an example illustrating the use of variance? ----CORRECT ANSWER-----All of these choices are true Which of the following conditions is needed regarding the chi-squared test statistic for the test of variance? ----CORRECT ANSWER-----All of these choices are true Under what condition(s) does the test statistic for p have an approximate normal distribution? ----CORRECT ANSWER-----When np and n(1 - p) are both > 5 In selecting the sample size to estimate the population proportion p, if we have no knowledge of even the approximate values of the sample proportion p̂ , we ---- CORRECT ANSWER-----Let p̂ = 0.50 When determining the sample size needed for a proportion for a given level of confidence and sampling error, the closer to 0.50 that p is estimated to be ---- CORRECT ANSWER-----The larger the sample size required Which of the following would be an appropriate null hypothesis? ----CORRECT ANSWER-----The population proportion is equal to 0.60 The analysis of variance is a procedure that allows statisticians to compare two or more population ----CORRECT ANSWER-----Means The distribution of the test statistic for analysis of variance is the ----CORRECT ANSWER-----F-distribution In one-way analysis of variance, between-treatments variation is measured by the ---- CORRECT ANSWER-----SST When is the Tukey multiple comparison method used? ----CORRECT ANSWER-----To test for differences in pairwise means In Fisher's least significant difference (LSD) multiple comparison method, the LSD value will be the same for all pairs of means if ----CORRECT ANSWER-----All sample sizes are the same Fisher's least significant difference (LSD) multiple comparison method is flawed because ----CORRECT ANSWER-----It will increase α; the probability of committing a Type I error When the objective is to compare more than two populations, the experimental design that is the counterpart of the matched pairs experiment is called a ----CORRECT ANSWER-----Randomized block design The primary interest of designing a randomized block experiment is to ----CORRECT ANSWER-----Reduce the within-treatments variation to more easily detect differences among the treatment means A complete 3 x 2 factorial experiment is called balanced if ----CORRECT ANSWER----- The number of replicates is the same for each of the 6 treatments In a two-factor ANOVA, there are 4 levels for factor A, 5 levels for factor B, and 3 observations for each combination of factor A and factor B levels. The number of treatments in this experiment equals ----CORRECT ANSWER-----20 A tabular presentation that shows the outcome for each decision alternative under the various states of nature is called a ----CORRECT ANSWER-----Payoff table Which of the following would be considered a state of nature for a business firm? ---- CORRECT ANSWER-----Worker safety laws A payoff table lists the monetary values for each possible combination of the ---- CORRECT ANSWER-----Event (state of nature) and act (alternative) Which of the following is true? ----CORRECT ANSWER-----All of these choices are true Which of the following statements is false regarding the expected monetary value (EMV)? ----CORRECT ANSWER-----In general, the expected monetary values represent possible payoffs Which of the following statements is correct? ----CORRECT ANSWER-----All of these choices are true The expected value of perfect information is the same as the ----CORRECT ANSWER-- ---Expected opportunity loss for the best alternative The expected value of sample information (EVSI) is the difference between ---- CORRECT ANSWER-----The expected monetary value with additional information (EMV') and the expected monetary value for the best decision (EMV*) The procedure for revising probabilities based upon additional information is referred to as ----CORRECT ANSWER-----Bayes' Law The difference between expected payoff under certainty and expected value of the best act without certainty is the ----CORRECT ANSWER-----Expected value of perfect information Which of the following is true about one-way analysis of variance? -------------------------- n1 = n2 = ... = nk is not required. the technique for hypothesis testing --------------------------concludes with either rejecting or not rejecting some hypothesis concerning a dimension of a population. in hypothesis testing the decision is based on the statistical evidence available costs (and profits) are only indirectly considered (in the selection of a significance level or in interpreting the p-value) in the formulation of a hypothesis test. In decision analysis --------------------------we deal with the problem of selecting one alternative from a list of several possible decisions. there may be no statistical data, or if there are data, the decision may depend only partly on them Decision analysis directly involves profits and losses. Because of these major differences, the topics covered previously that are required for an understanding of decision analysis are probability (including Bayes' Law) and expected value. payoff table --------------------------table showing the expected payoffs for each alternative in every possible state of nature Statistical inference problems involve three key concepts: -------------------------- population, the sample, and the statistical inference. Population: --------------------------the group of all items of interest to a statistics practitioner. It is frequently very large and may, in fact, be infinitely large. In the language of statistics, population does not necessarily refer to a group of people. It may, for example, refer to the population of diameters of ball bearings produced at a large plant. A descriptive measure of a population is called a parameter. In most applications of inferential statistics, the parameter represents the information we need. Sample --------------------------a set of data drawn from the population. A descriptive measure of a sample is called a statistic. We use statistics to make inferences about parameters. statistical inference --------------------------the process of making an estimate, prediction, or decision about a population based on sample data. Because populations are almost always very large, investigating each member of the population would be impractical and expensive. It is far easier and cheaper to take a sample from the population of interest and draw conclusions or make estimates about the population on the basis of information provided by the sample. However, such conclusions and estimates are not always going to be correct. For this reason, we build into the statistical inference a measure of reliability. There are two such measures, the confidence level and the significance level. The confidence level is the proportion of times that an estimating procedure will be correct. When the purpose of the statistical inference is to draw a conclusion about a population, the significance level measures how frequently the conclusion will be wrong in the long run. Statistical inference is the process of making an estimate, prediction, or decision about a population based on a sample. What can we infer about a Population's Parameters based on a Sample's Statistics? ---- ----------------------Since statistical inference involves using statistics to make inferences about parameters, we can make an estimate, prediction, or decision about a population based on sample data. We can apply what we know about a sample to the larger population from which it was drawn! Confidence level --------------------------the proportion of times that an estimating procedure will be correct. A confidence level of 95% means that estimates based on this form of statistical inference will be correct 95% of the time. significance level --------------------------measures how frequently the conclusion will be wrong in the long run. A 5% significance level means that, in the long run, this type of conclusion will be wrong 5% of the time. 𝛼 --------------------------Greek letter "alpha" If we use 𝛼 to represent significance, then our confidence level is 1−𝛼 Confidence Level + Significance Level = 1 Consider a statement from polling data you may hear about in the news: "This poll is considered accurate within 3.4 percentage points, 19 times out of 20." In this case, our confidence level is 95% (19/20 = 0.95), while our significance level is 5%. A 5% significance level means, that in the long run, this type of conclusion will be wrong 5% of the time. A company has developed a new smartphone whose average lifetime is unknown. In order to estimate this average, 200 smartphones are randomly selected from a large production line and tested; their average lifetime is found to be 5 years. The 200 smartphones represent a --------------------------sample Which of the following is a measure of the reliability of a statistical inference --------------- -----------a significance level The process of using sample statistics to draw conclusions about population parameters is called --------------------------doing inferential statistics Which of the following statements involve descriptive statistics as opposed to inferential statistics --------------------------The Alcohol, Tobacco and Firearms Department reported that Houston had 1,791 registered gun dealers in 1997. A population of all college applicants exists who have taken the SAT exam in the United States in the last year. A parameter of the population are --------------------------SAT scores Data --------------------------Facts and statistics collected together for reference or analysis three of the most popular methods to collect data: --------------------------direct observation (ex: number of customers entering a bank per hour), experiments (ex: new ways to produce things to minimize costs), and surveys. direct observation --------------------------The simplest method of obtaining data there are many drawbacks to direct observation. One of the most critical limitations of this data collection method is that it is difficult to produce useful information in a meaningful way. experiments --------------------------A more expensive but better way to produce data is through. Data produced in this manner are called experimental. survey --------------------------solicits information from people concerning such things as their income, family size, and opinions on various issues. The majority of surveys are conducted for private use. response rate --------------------------the proportion of all people who were selected who complete the survey. A low response rate can destroy the validity of any conclusion resulting from the statistical analysis. Statistics practitioners need to ensure that data are reliable. personal interview --------------------------involves an interviewer soliciting information from a respondent by asking prepared questions. A personal interview has the advantage of having a higher expected response rate than other methods of data collection. In addition, there will probably be fewer incorrect responses resulting from respondents misunderstanding some questions, because the interviewer can clarify misunderstandings when asked. But, the interviewer must also be careful not to say too much, for fear of biasing the response. The main disadvantage of personal interviews is that they are expensive, especially when travel is involved. telephone interview --------------------------usually less expensive, but it is also less personal and has a lower expected response rate. Unless the issue is of interest, many people will refuse to respond to telephone surveys. This problem is exacerbated by telemarketers trying to sell services or products. We can acquire information about the total population, make inferences within a stratum (gender), or make comparisons across strata (gender and age). Figure 2 outlines gender within a stratum. It highlights gender and age as an example comparison that can occur across data. mutually exclusive --------------------------means that each member of the population must be assigned to exactly one stratum. After the population has been stratified in this way, we can use simple random sampling to generate the complete sample cluster sample --------------------------a simple random sample of groups or clusters of elements versus a simple random sample of individual objects. Cluster sampling is particularly useful when it is difficult or costly to develop a complete list of the population members (making it difficult and costly to generate a simple random sample). It is also useful whenever the population elements are widely dispersed geographically. cluster sampling also increases sampling error because households belonging to the same cluster are likely to be similar in many aspects, including household income. This can be partially offset by using some of the cost savings to choose a larger sample than would be used for a simple random sample. Whichever type of sampling plan you select, you still have to decide what sample size to use. We can rely on our intuition which tells us that the larger the sample size is the more accurate we can expect the sample estimates to be. sampling error --------------------------refers to differences between the sample and the population that exists only because of the observations that happened to be selected for the sample. Sampling error is an error that we expect to occur when we make a statement about a population that is based only on the observations contained in a sample taken from the population. non-sampling error --------------------------more serious than sampling error because taking a larger sample won't diminish the size, or the possibility of occurrence, of this error. Even a census can (and probably will) contain non-sampling errors. Non-sampling errors result from mistakes that are made in the acquisition of data. Non-sampling errors also result from the sample observations being selected improperly. Three types of non-sampling errors are errors in: --------------------------data acquisition, non-response errors, and selection bias Data acquisition --------------------------errors arise from the recording of incorrect responses. Incorrect responses may be the result of incorrect measurements taken because of faulty equipment, mistakes made during transcription from primary sources, inaccurate recording of data due to misinterpretation of terms, or inaccurate responses to questions concerning sensitive issues such as sexual activity or possible tax evasion. Non-response error --------------------------refers to error (or bias) introduced when responses are not obtained from some members of the sample. When this happens, the sample observations that are collected may not be representative of the target population, resulting in biased results. The response rate: --------------------------the proportion of all people selected who complete the survey, is a key survey parameter and helps in the understanding of the validity of the survey and sources of non-response error. Non-response errors can occur for a number of reasons. An interviewer may be unable to contact a person listed in the sample or the sampled person may refuse to respond for some reason. In either case, responses are not obtained from a sampled person and bias is introduced. The problem of non-response error is even greater when self- administered questionnaires are used rather than an interviewer who can attempt to reduce the non-response rate by means of callbacks. Selection bias --------------------------occurs when the sampling plan is such that some members of the target population cannot possibly be selected for inclusion in the sample. Which of the following statements is true regarding the design of a good survey ----------- ---------------The questions should be kept as short as possible (part 2) Which of the following statements is true regarding the design of a good survey? --------------------------a. The questions should be kept as short as possible. b. A mixture of dichotomous, multiple-choice, and open-ended questions may be used. c. Leading questions must be avoided. d. All of these choices are true. Which method of data collection is involved when a researcher counts and records the number of students wearing backpacks on campus on a given day? -------------------------- Direct observation The manager of the customer service division of a major consumer electronics company is interested in determining whether the customers who have purchased a videocassette recorder over the past 12 months are satisfied with their products. If there are four different brands of videocassette recorders made by the company, the best sampling strategy would be to use a ________ --------------------------stratified random sample Which of the following types of samples is almost always biased? -------------------------- Self-selected samples A random experiment --------------------------an action or process that leads to one of several possible outcomes exhaustive --------------------------all possible outcomes must be included Additionally, the outcomes must be mutually exclusive, which means that no two outcomes can occur at the same time. A list of exhaustive and mutually exclusive outcomes is called a sample space and is denoted by S. The outcomes are denoted by 𝑂1, 𝑂2,....,𝑂𝑘 Using set notation we represent the sample space and its outcomes as: 𝑆={𝑂1,𝑂2,....,𝑂𝑘} There are three approaches to assign probability to outcomes. Each of these approaches must follow the two rules governing probabilities: --------------------------The probability of any outcome must lie between 0 and 1. That is:0 ≤ 𝑃(𝑂𝑖 )≤ 1 The sum of the probabilities of all the outcomes in a sample space must be 1. That is: Σ𝑘𝑖= 𝑃(𝑂𝑖) =1 classical approach --------------------------used by mathematicians to help determine probability associated with games of chance. If an experiment has n possible outcomes, this method would assign a probability of 1/n to each outcome. When rolling two dice, what is the total number of possible outcomes? ----------------------- ---there are 36 different outcomes but only 11 possible totals The conditional probability 𝑃(𝐴|𝐵)P(A|B) is called a posterior probability (or revised probability), because the prior probability is revised after the decision about taking the preparatory course. Identifying the Correct Method --------------------------Although it is difficult to offer strict rules on which probability method to use, nevertheless we can provide some general guidelines. The key issue is whether joint probabilities are provided or are required. Where the joint probabilities were given, we can compute marginal probabilities by adding across rows and down columns. We can use the joint and marginal probabilities to compute conditional probabilities, for which a formula is available. This allows us to determine whether the events described by the table are independent or dependent. We can also apply the addition rule to compute the probability that either of two events occurs. The first step in assigning probability is to create an exhaustive and mutually exclusive list of outcomes. The second step is to use the classical, relative frequency, or subjective approach and assign probability to the outcomes. There are a variety of methods available to compute the probability of other events. These methods include probability rules and trees. An important application of these rules is Bayes' Law, which allows us to compute conditional probabilities from other forms of probability. Bayes's Law is used to compute --------------------------posterior probabilities The classical approach describes a probability --------------------------in terms of the proportion of times that an event can be theoretically expected to occur If a set of events includes all the possible outcomes of an experiment, these events are considered to be --------------------------exhaustive Which of the following statements is not correct? --------------------------If event A does not occur, then its complement A' will also not occur. Sampling Distribution of the Mean --------------------------Sampling distributions describe the distributions of sample statistics. There are two ways to create a sampling distribution. The first is to actually draw samples of the same size from a population, calculate the statistic of interest, and then use descriptive techniques to learn more about the sampling distribution. The second method relies on the rules of probablility and the laws of expected value and variance to derive the sampling distribution. Central Limit Theorem --------------------------The theory that, as sample size increases, the distribution of sample means of size n, randomly selected, approaches a normal distribution. standard error of the proportion --------------------------the standard deviation of sample proportions, which measures the average variation around the mean of the sample proportions The concept that allows us to draw conclusions about the population based strictly on sample data without having any knowledge about the distribution of the underlying population is __________ --------------------------the central limit theorem Each of the following are characteristics of the sampling distribution of the mean except --------------------------if the original population is not normally distributed, the sampling distribution of the mean will also be approximately normal for large sample sizes Each of the following are characteristics of the sampling distribution of the mean: --------- -----------------the sampling distribution of the mean has a different mean from the original population the standard deviation of the sampling distribution of the mean is referred to as the standard deviation if the original population is not normally distributed, the sampling distribution of the mean will be normal Suppose you are given 3 numbers that relate to the number of people in a university student sample. The three numbers are 10, 20, and 30. If the standard deviation is 10, the standard error equals --------------------------5.77 You are tasked with finding the sample standard deviation. You are given 4 numbers. The numbers are 5, 10, 15, and 20. The sample standard deviation equals ----------------- ---------6.455 Two methods exist to create a sampling distribution. One involves using parallel samples from a population and the other is to use the --------------------------rules of probability hypothesis testing --------------------------make and test an educated guess about a problem/solution null hypothesis --------------------------a statement or idea that can be falsified, or proved wrong represented by 𝐻0 (pronounced H-nought) alternative or research hypothesis --------------------------the opposite of null hypothesis- consists of a statement about the expected relationship between the variables denoted 𝐻1 Type I Error --------------------------occurs when we reject a true null hypothesis. denoted by 𝛼, which is also called the significance level Type II Error --------------------------defined as not rejecting a false null hypothesis denoted by 𝛽 (Greek letter beta) The error probabilities 𝛼 and 𝛽 are inversely related, meaning that any attempt to reduce one will increase the other. Critical Concepts in Hypothesis Testing --------------------------1. There are two hypotheses. One is called the null hypothesis and the other the alternative or research hypothesis. 2. The testing procedure begins with the assumption that the null hypothesis is true. 3. The goal of the process is to determine whether there is enough evidence to infer that the alternative hypothesis is true. 4. There are two possible decisions:Conclude that there is enough evidence to support the alternative hypothesis.Conclude that there is not enough evidence to support the alternative hypothesis. test by stating that there is enough statistical evidence to infer that the null hypothesis is false and that the alternative hypothesis is true. if the value of the test statistic does not fall into the rejection region (or the p-value is large): --------------------------rather than say we accept the null hypothesis (which implies that we're stating that the null hypothesis is true), we state that we do not reject the null hypothesis, and we conclude that not enough evidence exists to show that the alternative hypothesis is true. Although it may appear to be the case, we are not being overly technical. One-Tail Test --------------------------Predicts that the results will fall in only one direction - either positive or negative Two-Tail Test --------------------------used when we want to test a research hypothesis that a parameter is not equal (≠) to some value Effects on 𝛽 of Changing 𝛼 --------------------------Decreasing the significance level 𝛼, increases the value of 𝛽 and vice versa. Shifting the critical value line to the right to decrease α will mean a larger area under the lower curve for 𝛽 and vice versa. The hypothesis of most interest to the researcher is --------------------------the alternative hypothesis A Type I error occurs when we --------------------------reject a true null hypothesis Statisticians can translate p-values into several descriptive terms. Suppose you typically reject H0 at level 0.05. Which of the following statements is incorrect? ----------------------- ---If the p-value < 0.01, there is overwhelming evidence to infer that the alternative hypothesis is false. In a criminal trial where the null hypothesis states that the defendant is innocent, a Type I error is made when --------------------------an innocent person is found guilty Population Mean --------------------------μ Population Proportion --------------------------P An unbiased estimator is --------------------------a sample statistic, which has an expected value equal to the value of the population parameter Thirty-six months were randomly sampled and the discount rate on new issues of 91- day Treasury Bills was collected. The sample mean is 4.76% and the standard deviation is 171.21. What is the unbiased estimate for the mean of the population? ------------------- -------4.76% A 98% confidence interval estimate for a population mean is determined to be 75.38 to 86.52. If the confidence level is reduced to 90%, the confidence interval for population mean --------------------------becomes narrower Suppose the population of blue whales is 8,000. Researchers are able to garnish a sample of oceanic movements from 100 blue whales from within this population. Thus, - -------------------------researchers can ignore the finite population correction factor In the sample proportion, represented by p = x / n, the variable x refers to ------------------- -------the number of successes in the sample analysis of variance --------------------------determines whether differences exist between population means. Ironically, the procedure works by analyzing the sample variance, hence the name. Analysis of variance is an extremely powerful and widely used procedure One-Way Analysis of Variance --------------------------analysis of variance in which there is only one grouping variable Independent samples are drawn from k populations: Note: These populations are referred to as treatments. It is not a requirement that 𝑛1=𝑛2=⋯=𝑛𝑘 𝑥 --------------------------the response variable, and its values are responses 𝑥𝑖𝑗 --------------------------refers to the ith observation in the jth sample (𝑥35 is the third observation of the fifth sample). The grand mean, 𝑥¯¯ --------------------------(read x-double bar), is the mean of all the observations where the sum of all observations is divided by the total of all observations from all samples: factor --------------------------Population classification criterion is called a factor. Each population is a factor level. Internet trading --------------------------has become quite common and online trades can cost as little as $7. It is now easier and cheaper to invest in the stock market than ever before Proportion of Total Assets Invested in Stocks --------------------------Percentage of total assets invested in the stock market is the response variable; the actual percentages are the responses in this example. Population classification criterion is called a factor. The age category is the factor we're interested in. This is the only factor under consideration (hence the term "one way" analysis of variance). You should confirm that the data are interval (percentage of total assets invested in the stock market) and that the problem objective is to compare four populations (age categories). The parameters are the four population means 𝜇1μ1, 𝜇2μ2, 𝜇3μ3, and 𝜇4μ4. The null hypothesis will state that there are no differences between the population means. Hence, 𝐻0:𝜇1=𝜇2=𝜇3=𝜇4 The analysis of variance determines whether there is enough statistical evidence to show that the null hypothesis is false. Consequently, the alternative hypothesis will always specify the following: 𝐻1:At least two means differ mean square --------------------------an estimate of either variance between groups or variance within groups F test (analysis of variance) --------------------------A statistical significance test for determining whether two or more means are significantly different. F is the ratio of systematic variance to error variance.