CNSL503/ CNSL 503 Module 3 Statistics 2026/2027 | Portage Learning | Verified Questions & Answers | 100% Correct | Grade A

CNSL503/ CNSL 503 Module 3 Statistics 2026/2027 | Portage Learning | Verified Questions & Answers | 100% Correct | Grade A Q: What is the difference between a statistic and a parameter? Answer A statistic is a descriptive statistical result that is generated from a sample, whereas a parameter is a statistical result from a population. Q: A ___ sample is composed of members that generally possess the same characteristics as those of the population. They allow for accurate inferences to be made. Answer representative sample Q: A ___ sample is not representative of the population. It is attributed to the researcher and data collection methods, and do not allow for accurate inferences to be made. Answer biased Q: This method of sampling is when every member of the population has an equal chance of being selected, this increases the reliability of results. Each member is independent of the others: selecting one member does not increase or decrease the likelihood of another member being selected. Answer Random sampling Q: This method of sampling is when each individual in the population of size has an equal chance of being selected for the sample, and relies on chance to create a representative sample. This is NOT feasible for large populations. Answer Simple random sampling Q: This method of sampling is a type of random sampling when populations can be subdivided into groups called strata. It randomly gathers data from subgroups within a sample. It is used when researchers wants to compare outcomes for different subgroups within a population or to compare outcomes within subgroups Answer Stratified sampling Q: Two types of stratified sampling: Answer 1. Proportional 2. Non-proportional Q: Proportional stratified sampling produces a sample that includes Answer similar proportions to the population Q: Non-proportional stratified sampling gathers equal sample sizes from each group regardless Answer of proportional stability Q: This method of sampling divides a population into subgroups known as clusters and then randomly selects several for the study. It is used when participants are geographically spread out as it is more efficient and cost effective Answer Cluster sampling Q: This method of sampling involves establishing a rule for how sample members will be selected (every 10th person, etc) Answer Systematic sampling Q: This method of sampling does not result in a representative sample as it involves selecting individuals based on them happening to be in a certain place, this can result in sampling bias Answer convenience sampling Q: The type of sampling chosen by a researcher is based on 3 things: Answer 1. nature of study 2. characteristics of the population 3. size of the sample needed Q: The nature of random sampling is that the sample statistics will deviate somewhat from the population parameters, resulting in Answer sampling error Q: A large enough ___ should be selected to represent the population to reduce sampling error. Answer sample size Q: An error that is not due to sampling, like due to data collection or measurement, is called Answer non-sampling error Q: ___ bias results from poorly worded/misleading questions or technical errors in a survey Answer Measurement bias Q: ___ bias results when participants respond inaccurately, untruthfully, or with exaggerated answers. Answer Response bias Q: __ bias is when those who participate are not truly a representative sample of the population Answer Selection bias Q: Biases are important to recognize because Answer they affect the result of the study Q: Researchers should report any known biases in a final study report so Answer readers can determine reliability Q: It is the responsibility of the researcher to ID sample size, select sampling technique, properly collect, analyze, and interpret data to Answer increase reliability of the results Q: It is the responsibility of the reader to critically evaluate claims of bias due to Answer the large quantity of data available for interpretation Q: A __ distribution graphs each value in a dataset Answer frequency Q: A ___ distribution shows a frequency distribution of each statistic from each possible sample os a given size, n from the population Answer sampling Q: This sampling statistic reports the frequency distribution of each mean from all possible samples of a given size n from the population. It takes on the shape of a normal curve as the number of samples in the study increases. As the number increases towards infinity, the sample mean approaches the population mean Answer Distribution of sample means Q: The closer the sample number is to the population number, the more representative the measures of descriptive statistics become and the less ___ is introduced Answer sampling error Q: ____ describes several characteristics of a distribution of sample means with three principles Answer central limit theorem The three principles of the central limit theorem: Answer 1. the mean of the distribution of sample means is equivalent to the population mean for large sample sizes (n= 30) 2. The distribution of sample means is an approximately normal distribution for large sample sizes 3. The standard deviation of the distribution of sample means is equal to the (population standard deviation/sample size of the distribution) The ___ is the standard deviation of the distribution of sample means, it indicates an amount of variability in a sample. Answer Standard error The formula for standard error is equal to Answer (population standard deviation/sample size of the distribution) A small standard error indicates Answer little variability, distribution is clustered near similar means A large standard error indicates Answer large variability, distribution is spread out 2 rules about computing standard error: Answer 1. If sample size increases, standard error decreases 2. If n=1, standard error is the same as population standard deviation In research, standard error is reported Answer in a sentence, then (sigma x-bar) or in a table using SE ___ describes the likelihood of a certain outcome occurring, it indicates what proportion of the whole is represented by a sample Answer Probability A __ is a possible result or observation Answer outcome A(n) __ is one or more outcomes that share a common aspect. Answer Event Probability (a) = (# of outcomes in a) / (total # of possible outcomes) Statistics are reported as ___ in stats decimals The ___ method is a way to calculate probability involving performing numerous observations of a given situation and recording the number of times an event occurs. Relative frequency method A ___ graph can be used to display the distribution of scores frequency distribution graph can be used to display the distribution of scores If sample statistics deviate greatly to a degree beyond chance, this deviation is considered ____. statistically significant. A p-value of __ is usually termed statistically significant. 0.05 The goal of _____ is to use information gathered from samples to form inferences and generalizations about a population. inferential statistics ___ is the formal process by which researchers sample data in order to make inferences about a population. The process by which is the claim is tested using a carefully selected sampling method, gathering data, calculating statistical values and probability values, and drawing a conclusion. Hypothesis testing Hypothesis testing is used to increase the reliability of claims 4 steps of hypothesis testing 1. establish the null and alternative hypothesis 2. establish criteria for rejecting the null hypothesis (setting a probability) 3. Collect the data and compute sample statistics 4. Decide whether to reject the null or fail to reject the null The __ hypothesis is the beginning assumption of any hypothesis test, stating that the treatment or independent variable had no effect, there is no statistical significance. Null (Ho) The __ hypothesis is that there is a change or difference in the population parameter, there is statistical significance alternative (Ha) The probability value used to determine rejecting or failing to reject the null hypothesis is stated in terms of a __ value alpha The alpha value is a small value that separates the distribution of sample means in 2 areas 1. area where high probability exists 2. area where low probability exists, the critical region. If sample lands in this region it provides evidence that the null hypothesis should be rejected A type __ error is a false positive type 1 error, researcher rejects the Null when they should fail to reject A type __ error is a false negative Type 2 error, researcher fails to reject when they should reject If a test statistic is greater than the critical value at the given alpha level, it can be determined statistically significant If the p-value is less than alpha, the null hypothesis is rejected Proven should/should not be used with statistical testing should not. Results either support or reject the null. Stats should be used to make inferences and generalizations and do not prove claims with 100% certainty. 4 reasons there is doubt with hypothesis testing and p-values 1. significance levels set arbitrarily 2. sample size greatly affects statistical significance 3. hypothesis testing should not replace valid, well-thought out scientific logical reasoning 4. Bias: cherry-picking statistically significant results, not publishing results that are not statistically significant Probability Range 0-1.0, closer to 1.0 for an event the greater likelihood the event will occur. 0.0 the lower the likelihood that event will occur. 0.5 indicates the likelihood the event happens is equal to the likelihood the event does not occur Experiment process or activity that generates observable well-defined otucomes Sample Point Each outcome of an experiment Sample Space The set of all possible outcomes an experiment may produce It is in form of a set and written as S= {list of sample points} ex: S={heads,tails} Probability distribution lists outcomes of a probability experiment with the probability for each outcome Probability Rules 1) probability for each outcome is between 0-1 inclusive Ei(single event) P(Ei) probability of an event 0 _ P(Ei) _ 1 for each i 2) sum of probabilities for all outcomes will equal 1 P(E1)+P(E2)...+P(En) = 1 Calculating Probability Classic Method & relative frequency method (empirical method) Classical Method for calculating probability Used when all outcomes of sample space are equally likely to occur Equally likely- n possibilities, 1/nth is the probability of occurring if chances are all the same ex S={1 2 3 4 5 6} n=6 P(1)=1/6, P(2)=1/6.. so on Relative Frequency Method/Empirical method of assigning probability Freq calculated by dividing numbers an event occurs divided by the total number of trials P(E)=#event E observed/occurs / #trials in experiment P(E)=N(E)/N(S) Counting Techniques (3 of em) Classical method for probabilities utilizes counting techniques 1. multiplication rule 2. combination rule 3. permutation rule Multiplication Rule Used for multi-step experiments ex flipping coin flipping coin twice gives us HH HT TH TT S={(H,H)(TH)(HT)(TT)} Multiplication rule, multi step. Sequence sequence with k steps with each having corresponding # nk, then total number given by total=n1n2n3.... Factorials Represented by ! n!=n(n-1)(n-2)....(2)(1) 0! = 1 ~special~ 7!= 765..=5040 3!=321=6 0!=1 Counting rule for combinations number of combinations is number of ways choosing r objects out of a set of n objects C(n,r)=(n!)/(r!(n-r)!) we count using combinations when we want to choose 'r' out of 'n' objects but we don't count all of the ways of ordering 'r' objects Counting Rule for Permutations number of ways of choosing AND ARRANGING 'r' objects from 'n' objects p(n,r) = (n!) / (n-r)! # permutations will always be larger than its combination counterpart bc we are counting ways of choosing and arranging r objects An event defined any subset of the sample space, sample space is possible outcomes. sample space s={1 2 3 4 5 6} event= E={2,4}, F={1,2,6} G={5} Complement of an event an event is defined as the collection of sample points making up an event, the complement is the collection of points in the space that ~do not~ make up an event. They are points left over from taking the sample points of the event away. defined as E^c= {sample set that is not in the event we are looking for) Complements and Probability notation A= event A Ac= complement of event A P (A)= probability of event A P (Ac)= probability of the complement of event A P(A)+P(Ac)=1 Addition Law Union of events and intersection of events Intersection of Events Intersection of two events, A and B, contains all sample points belonging to both A and B, intersection denoted as upside down U - in quizlet case 'n' AnB Union of events union of two events, A and B, contains all points belonging to A or B, or Both. Union denoted with 'U' AuB meaning union Addition Law provides means for calculating probabilities of Event A or B or both A and B occuring P (AuB) = P(A) + P(B) - P(AnB) Calculating probabilities using known probabilities P(AuB) = P(A)+P(B)-P(AnB) or P(AnB)= P(A)+P(B)-P(AuB) P(A)+(PAc)=1 or P(A)=1-P(Ac) rearranges variables allowing to work with #s given in a problem to algebraically find other probabilities Conditional Probabilities We determine event A has P(A) of occurring. Suppose we learn related event B has occurred and will influence probability of A. Denoted as P(A|B) P(A|B) "P event A will occur given that B has occurred" P(B|A) "P event B will occur given that A has occurred" Conditional Probability Equation P(A|B) = P(AnB) / P(B) or P(B|A) = P(AnB) / P(A) P(A|B) =/= P(A) if event A is dependent on B, then probability of A has occurred is going to be different than the probability that event A will occur (without B occurring) P(A|B) =/= P(A) A is dependent on B occurring, P(A|B)= P(A) If probability of event A occurring was not dependent on B, we expect probability of A occurring will be same whether or not B has occurred: P(A|B) = P(A) A is thus independent of the occurrence of B Test for independence Events are independent if and only if P(AnB) = P(A)*P(B) Bayes' Theorem The probability of an event occurring based upon other event probabilities. What is the difference between a parameter and a statistic? A parameter is a numerical value that describes a population, such as the population mean. A statistic is a numerical value that describes a sample, such as a sample mean A statistic is a descriptive statistical result that is generated from a sample, whereas a parameter is a statistical result from a population.. Why is it important to have a representative sample? It is important to have a representative sample, because we're trying to make inferences about a certain population. In order to make correct inferences about this population, the sample must reflect the population. For instance, not having a representative sample is what called Literary Digest to predict Alf Landon to win, when FDR actually won in a land slide. Representative sample consists of members that possess the same characteristics as those of the population (e.g. age distribution) biased sample a sample that is not representative of the population simple random sampling -every member of the population has an equal chance of being selected for the sample systematic sampling random sampling with a system in which the starting point is random and each subsequent member selected is based on a fixed interval (e.g. every 3rd person) - the probability that each member is selected is not independent of one another -random samples can be achieved in a way that is more efficient than simple random samlping stratified sampling is when you divide the population into strata (typically based on age, gender, race, socioeconomic status) and then you randomly select people to sample from each strata (not everyone in a strata is studied) -great for populations with subsections/ different characterstics cluster sampling is when you divide the population into clusters such as the U.S. voters into voters by states, and then you randomly select clusters (e.g. CA, NY, KY). From there, everyone in that cluster is studied. -great for very large populations convenience sampling selecting a sample that is convenient and easy to access - asking everyone in a lecture hall to determine L handedness at your school -does NOT result in a representative sample -generally is more prone to bias than other sampling methods Wha is sampling error ? Sampling error is the difference between the population parameter and the sample statistic. It is impossible to eliminate sampling error but there are ways to reduce it such as a larger sample size. How to reduce sampling error? Increase the sample size or use STRATIFIED sampling (increase the likelihood that the sample is more representative of the population) - the larger the sampling size the smaller the sampling error What are non-sampling errors? aka as? Examples? errors that are not the result of random sampling -sampling bias e.g. measurement bias, response bias, selection bias measurement bias may results from a mistake during the measurement process or poorly worded questions e.g. scale on carpet overestimates weight response bias when participants respond in a way that is inaccurate or untruthful selection bias when the sample is not representative of the population ( e.g. Literary Digest Alf Landon predicted to vote but the sample was upper class people who tend to vote republican) What is a sampling distribution? a frequency distribution of a statistic from every possible sample of a given size n from the population. What is a distribution of sample means? a frequency distribution of all possible sample means. Central Limit Theorem 1. The distribution of sample means is approximately normal as long as the sample size is large (n=30 or more) This is true REGARDLESS of the shape of the original population distrbiution 2. The mean of all the sample means is equal to the population mean! 3. The standard deviation of the distribution of sample means is known as the STANDARD ERROR and is calculated as the population standard deviation/ square root n standard error -Is the average difference between each sample mean and the population mean -indicates how much variability exists between samples -provides a measure of the SAMPLING ERROR (the larger the standard error the larger the sampling error) -We want the standard error to be small! How to reduce the standard error? Increase the sample size! (with a larger n, the standard error is smaller) probability The likelihood of a certain outcome occurring -must be between 0 and 1 (0= no chance, 1 = will always happen) -reported as a DECIMAL -may be described as "PROPORTION) -In research, probabilities are used to determine the likelihood that the result of a sample came from the original population. outcome Probability distributions often appear as.... the possible result of an observation histograms What is the probability that in a standard deck of 52 cards you get a heart? 13/52= 0.25 Statistical significance -A result is not likely to be from the population -For instance, the sample mean is so far in to the extreme that the probability is low that it could come from the original population By convention, if p is less than or equal to (p=.05) .05, the result is considered "statistically significant" It means that the probability that we got this sample mean or something more extreme from this population is 5% or less It's still possible but it is unlikely p-value P-value: the probability of obtaining the sample statistic (e.g. the mean) observed or more extreme from the original population By convention, if p is less than or equal to (p=.05) .05, the result is considered "statistically significant" relative frequency method involves performing numerous observations of a given situation and recording the number of times the event occurs - for finding probability of events where you are not 100% sure of the outcome Hypothesis testing the formal process by which sample data is used to evaluate a statistical hypothesis or a claim about a population (make inferences about a population) -based on probability (it's not black and white), making a best guess based on evidence from our sample What are the four steps of hypothesis testing? 1 State the hypotheses (Ho and Ha) Ho= There is no relationship between test scores and tutoring. (M= 75) Ha= There is a relationship between test scores and tutoring. (M=X 75, M75) 2. Establish the decision criteria (set the alpha level or significance level as 0.05) 3. Collect data and compute sample statistics as well as convert sample statistic into test statistic (e.g. z-score) 4. Make a decision (if p-value = alpha level, reject the null hypothesis, the result is statistically significant.) (if p value= alpha level, fail to reject the null hypothesis. there is no significant treatment effect). significance level the probability the researcher defines as "very unlikely" in a hypothesis test If a sample mean falls in the critical region, it is sufficiently unlikely to be the same as the untreated population what action? Reject Ho If a sample mean does not fall in the critical region, it is likely to be the same as the untreated population Fail to reject Ho T/F Hypothesis testing is Based on the logic of falsification T/F Hypothesis testing can prove claims. T. Based on the logic of falsification (trying to reject the null hypothesis) False Never ever say "prove" during hypothesis testing Combating Problems with Hypothesis Testing Supplement hypothesis testing with additional measures such as effect size and confidence intervals Problems with Hypothesis Testing 1. Conventional levels of alpha (.05, .01, or .001) are arbitrary We just somehow decided, but probabilities exist on a continuum 2. A larger sample size is more likely to achieve statistical significance than a smaller sample size Statistical significance does not always have meaningful significance Even small effects can be statistically significant with a large enough sample size 3 There is too much reliance on the p-value as the sole measure upon which conclusions are made Conclusions should also be drawn on past evidence, validity of assumptions made and so forth 4 People (including researchers!) misinterpret the p-value and the results of hypothesis testing 5 There is a bias towards publishing statistically significant results over non-significant results This leads to an incomplete and biased picture of the findings Alpha level (a)/ significance level the maximum probability that one would make a Type I error if the null hypothesis is true P-value the probability that we have made a type I error if the null hypothesis is true Critical values cutoff values, serve as the boundaries for the critical regions. If the test statistic lies in the critical region (region of low probability), then there is reason to reject the null hypothesis. The area in the distribution of sample means where a low probability exists critical region Which type of sampling involves a rule being established for how sample members will be selected? Systematic sampling Which of the following statements is NOT true regarding a representative sample? The results from a representative sample favor particular results. Which of the following is a disadvantage of simple random sampling? It is most practical for small populations. Sampling error is any deviation between a statistic and a parameter that occurs due to chance. Nonsampling errors usually result from problems with data collection. Examples of sources of nonsampling error include measurement bias, response bias, and selection bias. Sample size is considered large if it contains at least ____ members. 30 What theory identifies several key characteristics of a distribution of sample means? Central Limit Theorem What is another name for the standard deviation of a distribution of sample means? Standard error Calculate the standard error of a distribution of sample means for a sample size of 50 and a population standard deviation (σ) of 2.5. 2,5/ sqrt(5)= 0.35 What term refers to the likelihood of a certain outcome occurring? Probability What term is used to delineate a sample statistic that deviates from a population parameter to a degree that appears to be beyond chance? statistically significant What is the general value that is used to determine statistical significance? 0.05 Which of the following terms describes a claim that is made about a particular population parameter? hypothesis What is the difference between the null and alternative hypotheses? Describes some of the major problems with hypothesis testing. The significance level is an arbitrary value. - Hypothesis testing is sensitive to sample size. - P-values do not reflect the size of an effect - Hypothesis testing is often used as the only means to determine significance. - The results of hypothesis testing are frequently misinterpreted and misunderstood. - There is a publishing bias towards statistically significant results. What is the purpose of a significance level? The significance level (α) is the value that the P-value of a test statistic can be compared to in order to determine statistical significance. This value is typically 0.05. What is the difference between Type I and Type II errors? Type I errors are false positives, whereas Type II errors are false negatives. A type I error is a false positive. It is when you reject the null hypothesis when the null hypothesis is true. In other words, you are saying a treatment has an effect, when it is useless. It is the error that is more costly. A type II error is a false negative. It is when you fail to reject the null hypothesis when the null hypothesis is false. In other words, you are saying a treatment has no effect, when it is useful. This error is less costly. Identify the 4 steps in hypothesis testing. State the null (H0) and alternative (Ha) hypotheses. Establish the important cutoff (critical) values by which to measure the observed result to determine statistical significance. Calculate the test statistic. Compare the test statistic to the critical values, and decide if the null hypothesis (H0) should be rejecting or not. What is the difference between the null and alternative hypotheses? The null hypothesis (H0) is the beginning assumption of any hypothesis test that makes a particular claim about the value of a population parameter and, in essence, claims that an observed effect or difference does not exist. The alternative hypothesis (Ha) states that the value of the population parameter differs from the value claimed in the null hypothesis and, in essence, claims that an observed result or difference does exist.