Understanding Z-Scores and Sampling Distributions in Hypothesis Testing, Exams of Statistics

Download Understanding Z-Scores and Sampling Distributions in Hypothesis Testing and more Exams Statistics in PDF only on Docsity! Psych 308 - Psychological Statistics Exam 1 Which of the following is true regarding statistics? - Competence with statistics is a broadly applicable and highly marketable skill which is trending upward. Which of the following is the correct sequence involved in the scientific method? - Hypothesis formation>Research design>Data collection>Data analysis>Draw conclusions Which of the following is a difference between a population and a sample? - A population is all the people you are interested in, while the sample is all the people you have data on. Which of the following is ideal in a good sample? - representative of the population Which of the following best distinguishes constants and variables? - Constants are things that are the same across people; variables are things that vary across people. BYU professors and graduate students conducted a longitudinal study of links between time spent using social media and mental health (Coyne, Rogers, Zurcher, Stockdale, & Booth, 2020). Participants responded to a single social media use item, "How much time do you spend on social networking sites, like Facebook, on a typical day?," using a Likert scale from 1 (none) to 9 (more than 8 hours). How would you classify the social media use variable in terms of the nature of the phenomena and the scale of measurement? - Continuous; ordinal One frequently evaluated aspect of sexual risk-taking among teens is the number of sexual partners they have had. How would you classify the sexual partners variable in terms of the nature of phenomena and the scale of measurement? - Discrete; ratio Hundreds of studies have examined parenting using the parenting styles model, initially outlined by Dianna Baumrind. The four parenting styles are authoritative, authoritarian, permissive, and neglectful. Suppose you did such a study, assessed parenting, and then assigned each family to one of these four parenting styles. How would you classify the parenting styles variable in terms of the nature of the phenomena and the scale of measurement? - Categorical; nominal Which type of phenomena have magnitude (i.e., involve more or less of a phenomena)? - Discrete; continuous What is the key difference between nominal and ordinal scales of measurement? - Nominal does not have magnitude, ordinal does. What is the key difference between ordinal and ratio scales of measurement? - Ratio variables are more precise than ordinal variables. Which scale(s) of measurement can be used to operationalize a continuous variable (i.e., to turn it into data)? - Ordinal and ratio What scale of measurement is most ideal from a statistical standpoint? - Ratio What is one of the major pros of experimental research design? - Ability to infer causality b - Ordinal: This scale of measurement sets numbers on a scale of less or more, but not in a precise way. Each number on the scale means that there is an increase or decrease in amount, but it isn't clear how much of a difference there is between the two. An example of this could be the answer to the phrase "I am good with technology" (1-Strongly Agree, 2-Agree, 3-Neutral, 4-Disagree, 5-Strongly Disagree.) Each category shows an increase of agreeableness, but doesn't specify the amount more that the person agrees with the statement. c - Ratio: This scale of measurement does have set increments to the scale. Each number means less or more of something, and it also shows the precise difference in amount of each number on the scale. This means that you are able to do math on the thing, because number 2 is double the amount of number 1. An example of this could be how many cats you own (1, 2, 3, 4, etc.) You can't own half a cat, so this means that each level has an equal amount of increase. d - Nominal categorizes things by assigning a number. This means that there is no increase in value between number 1 and 2. In both ordinal and ratio, there is an increase in value between the two categories. The numbers in ordinal and ratio mean that there is less or more of something, where nominal just organizes them into categories with no numerical value placed on them. e - With ordinal, the increase in number/value is not specific. It shows some sort of incre Please (a) pick a variable you are interested in studying; (b) identify the type of phenomena it is— categorical, discrete, or continuous, and justify your answer; (c) identify and describe measurement approach you would use; (d) identifying the scale of measurement you would use—nominal, ordinal, or ratio, and justify your answer; (e) describe the way you would gather data—the methodology. - a - Something I am interested in studying is the amount of people who have different animal allergies, such as none, cat, dog, horse and bird. b - This type of phenomena would be categorical, because there isn't a numerical value between the different variables. c - I would gather my data using a survey format. Because this study is based on something that is already occurring in peoples lives, I would need to use a survey to determine the types of animal allergies people have. d - I would use the nominal scale of measurement because the categories wouldn't show in increase of value, there isn't less or more of something. It depicts the different categories, not an increased number. e - My study would be correlational, because it wouldn't have any sort of manipulation. I would also use a Cross-sectional study, because I wouldn't need to study the same group of people over a period of time in order to determine what kinds of animal allergies they have. I would also be using a real world setting, due to the fact that I would be observing the participants instead of manipulative a variable. This study explored individual, family, and peer predictors of involvement and psychological investment in fights among Samoan youth. Participants were 310 adolescents ages 13 through 19 living in Samoa, who completed a paper-pencil survey at school. MANCOVAs compared those involved in fights with those not, and those more invested in fighting with those less invested. In terms of individual predictors, for both of the independent variables, groups differed on school engagement, religiosity, proactive aggression, reactive aggression, moral disengagement, and social dominance but not self-regulation, empathy, or moral identity. Groups also differed on all peer variables: proactive aggression, reactive aggression, fight investment, and fight involvement. For involvement, groups differed on maternal and paternal psychological control and disrespect, but not parental monitoring. For investment, there were no differences - a - Population: The population in this study are Samoan youth. Sample: The sample in this study is 310 adolescents living in Samoa who were between the ages of 13 and 19. b - Constant: Everyone in the study were people who were living in Samoa during the time of the study. Variable: One variable would be religiosity of the students in the study. c - The study design would be correlational d - The study was Cross-sectional, because it was a one time survey, and it was also a real world setting, because the participants took the survey at school and it was based on their own lives. e - The study was conducted through a survey. Which of the following is a true compare/contrast between descriptive and inferential statistics? - Descriptive statistics describe the sample; inferential statistics make inferences about the population. Which of the following is a descriptive statistic? - mean Which of the following is an inferential statistic? - ANOVA Based on the data reported in the table below, at what percentile would you say someone would be who has dated 5 people? - 79th Which of the following best describes the shape of distribution shown above? - Highly positively skewed How many teens would be considered outliers? - 45 How would you describe the distribution shown above? - Normal In the graph above, about how many teens were 15? - 200 Which type(s) of variable(s) can be displayed using histograms? - Ordinal and Ratio What are the three characteristics of distributions? - Shape, central tendencies, and variability Participants in the NSYR dataset were asked "How often, if at all, do you think about or plan for your future?" Responses were on a 5-point Likert scale from 1 "Never" to 5 "Very often". Based on the content of the item, the response scale, and the descriptive statistics (M = 4.15, SD = .96), what might you anticipate the shape of the distribution to be? - Negatively skewed Which of the following variables is the most likely to be positively skewed? - Drug use Which of the following characteristics will likely lead to negative skewness? - High social desirability and/or low difficulty Which of the following is measure of central tendency? - tendency can be used to describe each and why, (d) explain for which variable(s) you cannot report indexes of variability and why, (e) explain for which variable(s) you can report indexes of variability and why. When discussing the "why" for each answer, speak in terms of scales of measurement - a - Gender is a nominal variable, and as a result, can only use the mode as a measure of central tendency. This allows you to be able to determine which gender is most frequent b - Education is used on a scale of 1-5, with each level depicting some sort of increase, due to the fact that they represent degrees in education. This means that education is an ordinal variable and can use mean, median, as well as mode within the data set. Mode would allow you to see which education is the most frequent, and the mean can tell you the average amount of education. The median will be able to tell you where the middle degree is, which can help determine skewness. However, it can only be used if the data set is an odd number or if the two middle numbers are the same. c - Income is coded in specific numbers, which is a dollar amount. This means that the scale of measurement used would be ratio, which means that you are able to use mean, median and mode for this variable. It would be best to use the mean for this in order to determine the average income for the sample, and the mode to determine which income is most frequent. The median can also be used to help determine the middle of the data set. d - We would not be able to report indexes of variability for the gender variable, because it is nominal, as well as the fact that it only has two categories within the variable. e - We are able to report indexes of variability for both the education and income variables due to the fact that they are ordinal and ratio, respectively. They use numerical scales, which can help us apply the data to help report indexes of variability. See the following formula: √(Σ(X-M)^2/N) . Please (a) give the name and symbol for this formula, (b) if we remove the square root, then what is the name and symbol for the formula, (c) what is the name and symbol for the numerator in the formula-- the top of the formula, (d) what is the name and symbol for the denominator--the bottom of the formula, (e) describe how values generated by this formula are interpreted—i.e., what do the numbers mean? - a - The formula shown above is used for Standard Deviation (SD), which can also be reported as σ, or "sigma." b - The name of the formula used without the square root would be the variance (SD^2). The variance can also be depicted as σ^2, or "sigma squared." c - The numerator in the formula is called the Sum of Squares (SS), which is also represented as the formula Σ(X-M)^2. d - The denominator, N, is known as the sample size in the data set. In other words, it is the number of values within the data. e - The standard deviation is used to help determine the average deviation from the mean. In other words, it shows us how far away, on average, a data point would be away from the mean while using the original units in the data set. It can tell us how far away a point is from the mean, by saying that is it "one standard deviation away from the mean." It can also help us determine any outliers, because outliers are generally 3 standard deviations away from the mean. Please write a results paragraph for the descriptive statistics shown above, using APA style reporting. Please report (a) the range of scores or minimum/maximum, (b) means, and (c) standard deviations. Also, (d) note any potential problems with the data, such as skewness, outliers, or missing data. Then (e) add any comments about what we learn about the phenomena from these descriptives. - In this study, 3370 religious youth were asked to report how often their church caused them to think of important things as well as how often their church was boring or disengaging by using a (a) 1- 4 (R=3) scale with 1 meaning usually, 2 meaning sometimes, 3 meaning rarely and 4 meaning never. In the first question, it was shown that 62% of youth answered 1, with those who answered the question with 2 were in the 90th percentile. On average the youth answered between a 1 and a 2 (b) (M=1.5). It is also important to note the standard deviation from the mean was at a .76 (c) (SD=.76). (d) Despite these results, there are also some potential issues with the data. However, out of the 3370 youth who were in the sample, 1162 of them did not submit their surveys. This can cause discrepancies within the data due to the fact that there were not as many scores to use as data. This survey also presented a slight positive skewness (Skewness=1.56). (e) This data shows that a majority of religious youth find themselves interested and engaged in their church, with only 3.31% of teens reporting that their church never caused them to think about important things. The results for the second question were quite different from the first. We asked the youth how often their church was boring or disengaging, while still using the (a) 1-4 scale (R=3). As opposed to the first question, only 14% of teens reported that their church was usually disengaging (1), with those who answered 2 being in the 51st percentile. For this question, the average answer was between a 2 and a 3 (b) (M=2.6) with the standard deviation from the mean at a 1.0 (c) (SD=1.0). The results for this question showed pretty normal results with little to no skewness (Skewness=.04) and similar scores on each scale. (d) There were also similar issues with this question as there What type of phenomena is region of the country? - Categorical What scale of measurement was used for region? - Nominal What percentage of BYU Student Study participants were from Utah - this is the variable utah (rounded; since this item was only asked of students from the USA, refer to the numbers in the Valid column not the Percent column)? - 33% What type of phenomena is attitudes about premarital sex? - Continuous What scale of measurement was used for premarital? - Ordinal What is the shape of the distribution for premarital (based on the skewness score)? - Moderately positively skewed What percentage of BYU Student Study participants thought that premarital sex is "always wrong"? - 64% What was the mean on premarital? - 1.58 What type of phenomena is time spent on scripture study? - Continuous What scale of measurement was used for scriptures? - Ratio What is the average amount of time spent on scriptures per day? - 13.51 minutes The z-score formula divides every deviation from the mean by the standard deviation of the scores. What is the purpose of subtracting the mean from each score? - Subtracting the mean changes the mean to 0. The z-score formula divides every deviation from the mean by the standard deviation of the scores. What is the purpose of dividing by standard deviation? - Dividing by the standard deviation changes the standard deviation to 1. Z-scores are in what units? - Standard deviations Which of the following is true of the z-distribution? - It has a mean of 0 and standard deviation of 1 The ACT has a population mean of 18 and standard deviation of 6. If a person has a z-score of 2, which of the following is true for them? - They have an ACT score of 30 The ACT has a population mean of 18 and standard deviation of 6. If a person has a z-score of -.5, which of the following is true for them? - They are a half standard deviations below the mean. About what percent of scores are beyond three standard deviations from the mean of a normal distribution (both directions combined)? - 1% The IQ has a mean of 100 with a standard deviation of 15 in the population. Let's say I randomly select 200 people from the population and have them take the IQ. Which of the following would be true? - If I randomly pick out one person from the sample, I have a 50% chance of getting someone with a score lower than 100 The IQ has a mean of 100 with a standard deviation of 15 in the population. If I randomly call a phone number in the U.S., and have the person who answers take the IQ, what are the chances they will scores between 85 and 115? - 68% Which of the following increases sampling error? - Larger population standard deviation What's the easiest way to decrease sampling error? - Get a larger sample. If we created a world without sampling error, and then drew samples from a population, and built a sampling distribution of means, what would the sampling distribution look like? - It would be a vertical line at the population mean. The mean of the sampling distribution of means will always be? - The population mean How do you get the mean and standard deviation of a distribution of means? - Use formulas based on the Central Limit Theorem to calculate the mean and standard deviation of the sampling distribution. Sampling distributions are always... - normally distributed Which pieces of information do you need to calculate the characteristics of the sampling distribution? - Population mean, population standard deviation, sample size What are my chances of drawing a mean from a population and getting a mean with a z-score greater than about 2 (plus or minus) on the sampling distribution? - About 5% The sampling distribution of means consists of ____. - all the possible means of samples of a given sample size drawn from a population Suppose you and your fiancé both run online studies collecting data on IQ, and you both end up with samples with M = 115. However, when you locate your sample mean on a sampling distribution it's in the outer 1%, while when your fiancé locates their mean on a sampling distribution, it is only in the outer 10%. What made the difference? - You had a smaller standard error. Means farther from the center in a sampling distribution of means... - will have a larger z-score and be less probable. A smaller z-score for a mean in a sampling distribution indicates... - a less extreme (more probable) mean Which of the following will get you a smaller z-score for a mean in a sampling distribution? - A sample mean closer to the population mean Population parameters - Population parameters are characteristics of a population, which are usually shown using the symbols μ, for population mean, and σ, for population standard deviation. For the most part, they are unknown, but they are also listed sometimes. Z-score - Z-scores are scores that show the location of raw scores on a distribution about, or compared to, the mean. They are listed in units of standard deviation. Sampling error - 1a. Your sample would be located 2 standard deviations away from the mean 2a. Roughly 2.5% of means would be above 2 standard deviations from the mean, and roughly 2.5% of means would be located below 2 standard deviations. This means that 5% of means would be more extreme than your mean. Which of the following is true regarding hypothesis testing? - Hypothesis testing is a part of inferential statistics, but not descriptive statistics. What is the official name of hypothesis testing? - Null Hypothesis Significance Testing Why do we directly evaluate the null hypothesis rather than the research hypothesis? - Because we can figure the probability of getting a particular result in the event that the null is true. How does the null hypothesis relate to the alternative hypothesis? - It's the absence of any effect (i.e., nothing is going on). What is the relationship between the research hypothesis and the alternative hypothesis for a single sample z-test? - The research hypothesis whichever you think is true out of the two alternative hypotheses and the null hypothesis. What is the formula for the general logic of hypothesis testing? - Our outcome / expected outcome given the null hypothesis Let's say we randomly draw 100 samples of 25 high school seniors and put their mean ACT scores in a pile (sort of a mini sampling distribution). Based on the central limit theorem, about 20% of those sample means should be above 19. Which of the following statements are true? - About 20 of the sample means will be above 19. What is the role of the sampling distribution in single sample z-tests? - You locate your mean in that distribution of means in order to determine how likely you would be to get it in the event that the null hypothesis were true in the population. Often the following options are used for possible cut-offs (alpha levels) for p-values: +p<.10, *p<.05, **p<.01, p<.001. Which one is used as a minimal criteria for calling a result "statistically significant"? In other words, your results have to at least be significant at this alpha level to be able to refer to them as "statistically significant." - *p<.05 When do we reject the null hypothesis? - When the probability of getting our sample mean if the null is correct is less than 5%. When do we retain the null hypothesis? - When the probability of getting our sample mean if the null is correct is greater than 5%. If our research hypothesis is aligned with one of the alternative hypotheses, and we retain the null hypothesis, what are the implications? - We failed to support the research hypothesis. If we reject the null hypothesis with a positive z-score, what is the most correct substantive interpretation? - Our sample is significantly higher than the null population. What does it mean to say your results are "statistically significant"? - You rejected the null in the hypothesis test. Let's say your research hypothesis was that you hypothesized that college students who listen to country music on their way to take the GRE will score lower than average. What if, heaven forbid, you run a single sample z-test and actually reject the null hypothesis, but in the direction of them scoring better than average. What should you conclude? - There is a statistically significant effect, it just isn't the one you expected. What does the z-formula tell me in a single sample z-test? - Ratio of the distance between my sample mean and the population mean over to the distance I might expect by chance if the null hypothesis is correct Which of the following is a true interpretation of z-scores in single sample z-tests? - If it is larger than 1.96, then your mean is significantly different from the mean of the null hypothesis population. Which of the following will get you a larger z-score for a mean in a z-test? - A sample mean farther from the population mean What does the following formula tell us? (M-μM)/σM - It is the z-score formula for locating an M in a distribution of Ms. How should you interpret a z-score of 2.0 in a z-test? - Your sample mean is two standard deviations above the mean of the null population. If I have a z-score of 2 in a z-test, about what percent of means in the distribution of means are more extreme (in both directions combined) than my sample mean? - 5.% How should we interpret study results if p=.023? - There is a 2.3% chance of getting our sample mean if the null is true, so we reject the null. How should we interpret study results if p=.065? - There is a 6.5% chance of getting our sample mean if the null is true, so we retain the null. If you reject the null with a single sample z-test, and it is in the direction you expected, what language should you use: - Which of the following is a benefit of effect sizes? - They are not influenced by sample size. Which of the following is NOT one of the three main things we want to know about effects, from a statistical standpoint? - Whether or not the effect aligns with theory. What's the difference between the z-test and Cohen's d? - The z-test is interpreted in terms of statistical significance, while Cohen's d is interpreted in terms of size. Which of the following is true regarding how hypothesis tests and effect sizes work together? - They supplement each other, so you always want to report both. Which of the following is a characteristic of the Cohen's d effect size? - c. It is the distance between two population means in standard deviation units. What's the difference between the z-test formula and the Cohen's d formula? - The standard deviation on the bottom of the formula What is the point of dividing the difference between population means by the population standard deviation? Why not just use the raw distance between population means as the effect size? - It standardizes the difference between population means (i.e., it puts them all on the same scale, which is standard deviations). Where does the mean to the left of the negative sign come from (µH1) in the Cohen's d formula associated with a single sample z-test? - It is the mean of the alternative hypothesis population, which is estimated by the mean of the sample. In what units is Cohen's d? - Standard deviations What do you assume when interpreting an effect size? - The alternative hypothesis is true. On what distribution is the effect size? - Population How would you interpret the results of the following results: z = 1, d= 1? - Retain the null with large effect size. Which of the following would be considered a large Cohen's d? - .75 What is the interpretation of a Cohen's d of 2.00? - The alternative hypothesis population mean is two standard deviations above the null hypothesis population mean. Which of the following leads to a larger Cohen's d effect size? - Larger difference between sample and population mean When we are not given population parameters, what do we do? - We estimate them with sample data but know there is likely some error. What point estimate (estimate of a population parameter) are we usually most interested in with a single sample z-test? - The mean of the alternative hypothesis population. On which distribution are confidence intervals based? In other words, where do we work out the calculations and visualize what's going on? - The alternative hypothesis sampling distribution. What is an incorrect interpretation of a 95% CI? - There is a 95% chance that the true population parameter value lies within that interval. What is the correct interpretation of a 95% CI? - With repeated sampling, 95% of the time the true population parameter value will lie within that interval. If you reject the null hypothesis in a z-test of means, which of the following is true regarding the confidence interval? - The 95% confidence interval does not include the null population mean. The confidence interval is centered around what? - The point estimate of the population parameter Which of the following is NOT used in calculating the confidence interval? - Effect size Which of the following will typically correspond perfectly with what you learn from the confidence intervals? - Hypothesis testing results Which of the following is preferred? - A narrow confidence interval Effect size (more generally, not just d) - Indexes for capturing the size of effects. It tells you how big/what size the effect is. Smaller population standard deviation When is the best time to calculate power? - Before conducting your study What's the typical amount of power we shoot for in psychology? - 80% If you retain the null, but you have a very large effect size, what should your first hunch be about what is going on? - Power might be low due to a small sample size. Which of the following assume the null hypothesis is true? - Hypothesis test On which distribution can power (and Type II error) be visualized? - On the sampling distribution derived from the alternative hypothesis population. What is the link between beta (Type II error) and power? - They add up to 1 (or 100%) Which of the following gives you the most power? - A large sample size and large effect size. Which of the following does NOT influence power? - Your hypothesis testing results Which of the following critical values gives you the most power? - +p < .10 (Zcrit= 1.28) Which of the following is true regarding power? - One-tailed single sample z-tests have more power than two-tailed single sample z-tests. Which of the following is the most legitimate way to increase your power? - Get a larger sample size. Which question do hypothesis tests address? - Is there an effect? Which question do effect sizes address? - If there is an effect, how big is it? Which question do confidence intervals address? - How precise is our estimate of the effect? Which question does power address? - What is our likelihood of finding the effect, if it is there? Which of the following is true regarding Type I error? - It corresponds to your critical p-value. Which of the following is true regarding Type II error? - 1-Power is your probability of a Type II error. Which of the following is true regarding Type I and Type II error? By "assume" we mean what has to be true in the population in order for it to be an error. - Type I error assumes the null hypothesis is true, Type II error assumes the alternative hypothesis is true. What's the typical alpha level (Type I error rate) used in psychology? - 5% What's the typical beta level (Type II error rate) used in psychology? - 20% If I use the standard alpha level of .05, and I conduct 100 studies where the null hypothesis is correct, how many of them are likely to be Type I errors (false rejects)? - 5 If I use the standard beta level of .20, and I conduct 100 studies where the alternative hypothesis is correct, how many of them are likely to be Type II errors (false retains)? - 20 If we set the alpha level for a hypothesis test at 0.01 instead of 0.05, which of the following is true? - We have a lower risk of a Type I error (false alarm). If we decrease our Type I error rate (e.g., .01 rather than .05), what will happen to our Type II error rate? - It will increase Hypothesis testing - The process of figuring out the probability of getting our results by chance, if the null hypothesis is true. It is the probability of getting what we got if the null is true. Power - The probability of rejecting the null if the alternative hypothesis is true. In other words, it's the probability of a "correct reject" which is determined by when the power level is greater than .80, or 80%. See notebook, Quiz 6 What is the difference between the sample mean and the mean of the null population? - .48 What is the standard error for the z-test comparing the sample mean to the population mean? - .01 What is the average amount you expect sample means to err from the null population mean when sampling samples of N = 933? - .01 (standard error) What is the average distance means in the sampling distribution of means are from the null population mean? - .01 (Standard error) How many standard deviations away from the null population mean is your sample mean in the sampling distribution of means? - 49.30 (Z score) What is the ratio of the size of the difference between your mean and the null hypothesis population mean you actually got compared to what you might expect to get due to sampling error? - 49.30 What is the z-score for the z-test comparing the sample mean to the population mean? - 49.30 What is the p-value for the z-test comparing the sample mean to the population mean? - <.001 What is the likelihood of you randomly sampling N = 933 people with a mean of 3.88 from the null hypothesis population? - <.001 What is the likelihood of you randomly sampling N = 933 people with a mean of 3.88 if the null hypothesis is true? - <.001 What level of significance did your results meet? - *** p <.001 Which of the following is true? - Your z-score was larger than the z-critical Your p-value was smaller than the alpha level Both What is the statistical decision? - Reject the null What is your estimate of the population mean of the alternative hypothesis population? - 3.88 What is the 95% CI for your estimate of the alternative hypothesis population mean? - 3.86, 3.90 Which of the following is true? - Your p-value was less than .05, and your 95% confidence interval did not include the null population mean. What is the Cohen's d effect size? - 1.60 How many standard deviations away is the alternative hypothesis population mean from the null hypothesis population mean? - 1.60 (Cohen's d) How would you evaluate the size of this effect size? - Large What is your basic conclusion of this single sample z-test? - Supported research hypothesis, and large effect. Write up these results using APA style. Please (a) write up the descriptive statistics—reporting the sample size, mean, and standard deviation, and commenting on the shape of the distribution (and whether it is approximately normal or is substantially skewed). Next, (b) report the hypothesis testing results, (c) effect size, (d) and confidence interval. Lastly, (e) write the substantive conclusions from the analysis. This all should be formally as if you were writing this up for a peer-reviewed journal. - We conducted a single sample z-test to compare the average high school gpa of BYU students to the average gpa of US students. Our hypothesis was that, on average, BYU students are smarter than the average high school student in terms of high school gpa. The single sample z-test showed that BYU students had a significantly higher gpa, z = 49.30, ***p < .001. Specifically, the average gpa for BYU students was M = 3.88, 95% CI [3.86, 3,90], compared to the population (μ = 3.4, σ = .3), and there was a large effect (d = 1.6) Great job overall! Don't forget to read the whole question - we wanted to see something about the shape of the distribution, as well as e - substantive conclusions. Great job through!