Final Exam practice (4)

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Studocu is not sponsored or endorsed by any college or university Final Exam practice Introduction to Statistics (Queen's University) Studocu is not sponsored or endorsed by any college or university Final Exam practice Introduction to Statistics (Queen's University) Downloaded by Kirsten Schneider (k.schneider4696@gmail.com) lOMoARcPSD|13275606

1 MULTIPLE CHOICE – Write your answer in the space provided, using UPPERCASE letters. Illegible answers will be marked incorrect. You DO NOT need to justify your answer. 1) When doing a significance test, a student gets a p-value of 0.06. This means: [1 mark] I. Assuming Ho is true, the sample results are a likely event. II. 94% of samples should give results that fall in this interval. III. We reject Ho. A. I only B. III only C. I and II D. I and III E. II and III ANSWER: ____A______ 2) What is the purpose of inferential statistics versus descriptive statistics? [1 mark] A. Inferential statistics describe the outcome of random phenomena B. Inferential statistics are used to make informed predictions about parameters we don’t know C. Inferential statistics describe samples that are normally distributed and large enough (n>30) D. Inferential statistics are used to generate samples of random data for a more reliable analysis E. Inferential statistics are used to predict outcomes for games of chance ANSWER: _____B_____ STUDY A (questions 3 & 4) You are interested in the flowering time of tulips in early spring and decided to keep track of the date that different gardens in your neighborhood bloom. You use April 1 st as a reference point and record the number of days past this date for each garden. You sample 13 gardens and find that the mean number of days is 12.3 with a standard deviation of 3.1. 3) For STUDY A, what is the 95% confidence interval for the average days past April 1 st that the gardens bloomed based on this sample? [1 mark] A. (10.4,14.2) B. (5.1,18.1) C. (5.5,19.1) D. (-1.9,1.9) E. (10.7,13.8) ANSWER: ____A______ Downloaded by Kirsten Schneider (k.schneider4696@gmail.com) lOMoARcPSD|13275606

2 4) For STUDY A, which of the following statements about the confidence interval is true? [1 mark] A. The confidence interval indicates the range over which you would fail to reject the null hypothesis of a single-sample t-test. B. The average number of days before flowering in the sample is between the days given by the interval. C. There is a 95% probability that the true average number of days before flowering is between the days given by the interval. D. The confidence interval is based on a z-distribution E. None of the above. ANSWER: ____B______ STUDY B (questions 5, 6 & 7) As a mushroom farmer in the Kingston region, you are trying to find new opportunities to sell the range of goods you produce. You decide to try developing homegrown mushroom kits, but wanted to do a market survey first to see if the average homeowner could growth mushrooms successfully. You approach 23 random people at the weekend farmers market and ask if they would be willing to try growing Oyster mushrooms from your kit. You first ask them whether they have previous experience growing mushrooms and then follow up with each participant each week over the next three weeks and ask them whether the mushrooms successfully grew. You are interested in evaluating whether success at growing mushrooms by the end of the 3 weeks depended on previous experience. 5) For STUDY B, what statistical test would be most appropriate for testing your question? [1 mark] A. One-sample t-test B. Two-sample t-test C. Chi-square test D. Single-factor ANOVA E. Two-factor ANOVA ANSWER: ____C______ 6) For STUDY B, identify the type of study design. [1 mark] A. Cohort study B. Stratified survey C. Case-control study D. Cluster survey E. Simple random survey ANSWER: ____A______ Downloaded by Kirsten Schneider (k.schneider4696@gmail.com) lOMoARcPSD|13275606 C

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3 7) For STUDY B, select the plot that is most appropriate for showing whether success at growing mushrooms by the end of the 3 weeks depended on previous experience. [1 mark] A. Bar chart B. Box plot C. Grouped bar chart D. Grouped box chart E. Scatter plot ANSWER: _____C_____ 8) Which of the following statements is FALSE? [1 mark] A. We repeatedly sample a population in practice to create the sampling distribution B. The ‘true’ sampling distribution does not change among samples C. The population distribution does not change among samples D. The sample mean is variable among different samples E. The estimated standard error is variable among different samples ANSWER: _____A_____ 9) Which of the following statements best explains why confidence intervals cannot be used to test for significance in a t-test? [1 mark] A. Confidence intervals do not share a common sampling distribution with t- tests. B. Confidence intervals indicate the range over which the true population value lies. C. Confidence intervals are a statement about the sampling distribution of your population, not of the null distribution. D. Confidence intervals do not use the standard error estimated from your sample. E. Confidence intervals are always two-tailed whereas t-test can be one- or two- tailed. ANSWER: _____C____ Downloaded by Kirsten Schneider (k.schneider4696@gmail.com) lOMoARcPSD|13275606

4 STUDY C (questions 10-13) You discover a pamphlet advertising final exam prep sessions for statistics. Since exams are coming soon, you decide to read what they have to say. Use your statistical knowledge to find the statistical and methodological errors. Improve your stats grade by 50%! (entirely fictional) Our company (StatsHeroes) has spent the last 3 years developing the most efficient exam prep material to help you excel in your statistics course. We developed the material based on a survey of students who had previously taken a statistic courses. We selected 100 students who had the top grades in their course, and asked them a series of questions about exam prep strategies. Using a linear regression, we found that students who used YouTube videos versus taking notes from the lectures as a study method performed statistically better on the exams. Our analysis was done using a t-test on the slope and we rejected the null hypothesis (t_observed=0.3, t_critical=2.1). We found that students who slept well the night before an exam felt less stressed heading into the exam (Chi-square test, p<0.05). Working with our sample of students, we developed a series of practice questions as part of the prep material that are designed to improve your grade. To test the effectiveness of these practice questions, we asked all of the students in our sample to take an exam, then let them use the practice questions to hone their statistical skills, and finally asked the same students to retake the exam. We analyzed the results using a two- sample t-test and found a significant result (p=0.012). The p-value indicates that our practice questions are guaranteed to improve your score 99.8% of the time! 10) For STUDY C, which of the following is the most important methodological error found in the passage? (Multiple answers possible: 1 mark for each correct response. You will lose 0.5 marks for each incorrect answer, but only applied to this question.) A. The study design has pseudoreplicated sampling units B. The study design has biased sampling unit allocation C. The study design uses non-random sampling units D. The study design is missing a control level needed to compare. E. There are no methodical errors ANSWER: ____C_____ 11) For STUDY C, which of the following is an error found in the passage? [1 mark] I. Testing the study methods should not be done using linear regression. II. Linear regression is not evaluated using a t-test. III. There is a mismatch between the test scores and the statistical conclusion. A. I only B. III only C. I and II D. I and III E. II and III ANSWER: ____D______ Downloaded by Kirsten Schneider (k.schneider4696@gmail.com) lOMoARcPSD|13275606

5 12) For STUDY C, which of the following is an error found in the passage? [1 mark] I. Evaluating the relationship between sleep and stress should not be done with a Chi-square test. II. Evaluating the effectiveness of the practice questions on exam success should not be done using a two-sample t-test. III. The researchers have drawn an incorrect scientific conclusion about the p- value for the effectiveness of the practice questions on exam success. A. I only B. III only C. I and II D. I and III E. II and III ANSWER: ____E______ STUDY D Use the following study to answer questions 13, 14 & 15. Airbnb is an accommodation-sharing platform that relies heavily on guest ratings. Interestingly, the distribution of ratings (shown below) is not equal across the possible scores and is also not Normally distributed (5.0 is excellent, 1.0 is poor). The proportion of accommodations in each category is: Rank 5.0 is 46.3% Rank 4.5 is 41.1% Rank 4.0 is 10.3% Rank 3.5 is 1.7% Rank 3.0 is 0.3% Rank 2.5 is 0.2% Rank 2.0 is 0.1% Rank 1.5 is 0% Rank 1.0 is 0% 13) If you took repeated samples of potential accommodations for STUDY D, which of the following statements about the distribution of the mean ratings would be INCORRECT? [1 mark] A. Since the data are not Normally distributed, the distribution of mean ratings cannot be Normally distributed. B. The mean of the distribution is the same as the mean of the population distribution. C. The variance of the distribution depends on sample size. D. Sampling error causes the variation in the distribution. E. The standard deviation of the distribution can be estimated from a sample. ANSWER: ____A_____ Downloaded by Kirsten Schneider (k.schneider4696@gmail.com) lOMoARcPSD|13275606

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6 14) You collect a sample of 32 accommodations for STUDY D and find the following information about the ratings: Mean=4.2, Median=4.5, SD=0.2, IQR=1.5. Based on this information, what is the estimated standard deviation of the distribution of mean ratings? [1 mark] A. 0 ≤ ANSWER < 0.05 B. 0.05 ≤ ANSWER < 0.1 C. 0.1 ≤ ANSWER < 0.2 D. 0.2 ≤ ANSWER < 0.5 E. 0.5 ≤ ANSWER < 1.0 ANSWER: _____A_____ 15) What is the probability that a randomly selected accommodation in STUDY D will have a rating that is less than 4.0? [1 mark] A. 0 ≤ ANSWER < 0.2 B. 0.2 ≤ ANSWER < 0.4 C. 0.4 ≤ ANSWER < 0.6 D. 0.6 ≤ ANSWER < 0.8 E. 0.8 ≤ ANSWER < 1.0 ANSWER: _____A_____ 16) You have been investigating the human health effects of a factory that has been releasing contaminated water into a river from which all local communities draw their drinking water. The toxin in question is known to cause liver cancer in lab rats. To determine whether pollution in the river is affecting human health, you randomly sample 200 people living in the community upstream from the factor (that don’t drink the polluted water) and 200 individuals that live downstream (hence they do drink the contaminated water. For each person sampled, you perform a liver enzyme test to look for the tell tale signs of liver cancer. What type of statistical test would be best suited to analyze these data? [1 mark] A. Chi-square test B. Paired-sample t-test C. 2-sample t-test D. 1-factor analysis of variance (ANOVA) E. Regression ANSWER: _____C_____ Downloaded by Kirsten Schneider (k.schneider4696@gmail.com) lOMoARcPSD|13275606

7 17) You want to plant a butterfly garden on the balcony of your apartment, but are not sure what species of milkweed is best suited for pots. You grow 10 plants of the ‘Common Milkweed’ and 10 plants of a related species called the ‘Butterfly Weed’ and measure their height after 2 months. Which of the following statements is about the descriptive statistics for your data? [1 mark] I. A boxplot showing that the interquartile range for the common milkweed is greater than for the butterfly weed. II. The difference in median height between the two species is 3.2 cm. III. The height difference between the two species is unexpected from sampling error. A. I only. B. II only. C. I and II. D. I and III. E. II and III. ANSWER: _____C_____ 18) Which of the following questions should be analyzed using a 1-tailed t-test? [1 mark] A. Is the fluorine concentration in Kingston’s water above the recommended guide of 0.7 mg/L? B. Does the mean patient wait time differ between an After-Hours health clinic compared to a regular doctor’s office? C. Has the first calendar date that stores put out their Christmas ornaments for sale changed over the past 20 years? D. Is the amount of sea ice in our Arctic waters different now from what it was 10 years ago? ANSWER: _____A_____ 19) Sara is an ornithologist and has been studying stress hormones (ug/ml) in black- capped chickadees over the winter months. She collects blood samples from 10 random birds in a forest site, 10 random birds from a field site, and 10 random birds from a urban site. What statistical test should be used to evaluate whether there is a difference in stress hormones among the sites? [1 mark] A. Chi-square test B. Paired-sample t-test C. 2-sample t-test D. 1-factor analysis of variance (ANOVA) E. Regression ANSWER: _____D_____ Downloaded by Kirsten Schneider (k.schneider4696@gmail.com) lOMoARcPSD|13275606

8 20) The following R output is for a linear regression of brain mass in bats as a function of their body mass. Which of the following values is used to test the hypothesis that body mass can be used to predict brain mass? [1 mark] A. The estimate value of 3.97821. B. The t-value value of 44.44. C. The multiple R-squared value of 0.9272 D. The t-value value of 27.88. E. The estimate value of 0.75164 ANSWER: _____D_____ 21) You have been given a dataset that contains the viral load for patients who are sick with the flu. The dataset also includes whether the patients had been given a flu vaccination (levels of yes/no) and their age group (levels of child/teen/adult/senior). Which of the following figures would best illustrate whether the efficacy of the flu vaccine for reducing viral load depends on patient age? [1 mark] A. Boxplot B. Contingency table C. Interaction plot D. Histogram E. Scatter plot ANSWER: _____C_____ 22) Select which of the following null and alternative hypotheses are most appropriate for a two-sample t-test that answers the following question: “Are the means of my samples different?”. [1 mark] A. H 0 : μ ≤ 0 H A : μ > 0 B. H 0 : μ A = μ B H A : μ A ≠ μ B C. H 0 : μ A ≥ μ B H A : μ A < μ B D. H 0 : μ > 0 H A : μ ≤ 0 E. H 0 : μ A ≠ μ B H A : μ A = μ B ANSWER: ____B_______ Downloaded by Kirsten Schneider (k.schneider4696@gmail.com) lOMoARcPSD|13275606

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9 SHORT ANSWER - Write all your answers in the space provided. 23) Carbon tax is a hot political issue right now in Canada. The idea is that by taxing activities or products that generate a lot of carbon dioxide, consumers will change their behavior and opt for choices that generate less carbon dioxide. You have been asked to conduct a survey looking at how people feel about carbon taxes in five key areas: gasoline, cement manufacturing, electricity generation from natural gas plants, home heating with fossil fuels, agriculture. You decide to run the survey online by sending emails to 1000 people who have their email registered with Revenue Canada. The survey asks participants to select their level of support (strongly support, support, neutral, do not support, strongly do not support) for carbon taxes under each area. A. Indicate the sampling unit and the statistical population (be specific). [2 marks] [sampling unit: a person (or person’s email); statistical population: all the people (or emails) who have an email registered with Revenue Canada. (Zero marks for not identifying that the statistical population is just those that have an email registered with revenue Canada.)] B. Give one example of a hidden bias that could arise from this sampling method (be specific). [1 mark] [People without an email registered with Revenue Canada will not be included in the sample, such as those who are older or who can’t afford a computer. These people are likely to have a different perspective on carbon tax, which may cause a bias in your answer] C. Indicate the most appropriate statistical test (be as specific as possible) [1 mark] [Chi-square test] D. Explain why you selected this test [1 mark] [There are two categorical variables, which can only be analyzed with a Chi- square test.] E. Indicate the null and alternative hypothesis (be mindful of direction in the test if appropriate) [1 mark] [The null hypothesis is H0: observed counts are not different from the expected counts, HA: observed counts are different from the expected counts. *grade part c based on the answer in part a, even if that was incorrect] F. Name the appropriate test statistic (e.g., F-score) (You do not need to find its value.) [1 mark] [Chi-square score. *grade part d based on the answer in part a, even if that was incorrect] Downloaded by Kirsten Schneider (k.schneider4696@gmail.com) lOMoARcPSD|13275606

10 24) You work for a cosmetic company that develops tanning solutions designed to modify skin color. You have been asked to evaluate the effectiveness of two formulations (Dihhydroxyacetone, Erythrulose) and three methods of application (Cream, Gel, Solution) in terms of color change. After running the experiments and collecting the data, you analyze the data and generate the following R output. Use these results to answer the following questions. Downloaded by Kirsten Schneider (k.schneider4696@gmail.com) lOMoARcPSD|13275606

11 24) continued… A. Indicate the most appropriate statistical test (e.g., t-test, regression etc.) for this data and explain your rationale. Be as specific as possible. [1 mark] This is a two-factor ANOVA [0.5 marks], which is most appropriate because we are studying a numerical response under two categorical explanatory variables [0.5 marks]. B. Indicate the statistical distribution you will use to test the null hypothesis. [1 mark] It is an F-distribution. C. Your employers are interested in whether the color change for each formulation depends on the type of application. For this specific question, state: a. The null and alternative hypothesis. [1 mark] δ F1A1 =δ F2A1 =δ F1A2 =δ F2A2 =δ F1A3 =δ F2A3 = 0, where delta is the difference of each cell from additivity. b. The observed test score (3 significant figures). [1 mark] F_observed=4.677 c. The appropriate degrees of freedom and the associated critical test score (3 significant figures) from the table provided at the end of the exam. Assume a Type I error rate of 5%. [1 mark] Numerator degrees of freedom is 2, denominator degrees of freedom is 18. From the table, this gives a F_crit=3.555 d. Your statistical conclusion. [1 mark] Since F_crit<F_obs, we reject the null hypothesis (0.5 marks) and conclude that some of the cells are different from additivity (0.5 marks). Downloaded by Kirsten Schneider (k.schneider4696@gmail.com) lOMoARcPSD|13275606

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12 24) continued… D. The following boxplot shows the results of the experiment. Labels are ‘D’ for Dihhydroxyacetone, ‘E’ for Erythrulose, ‘Cream’ for cream application, ‘Gel’ for gel application, and ‘Sol’ for solution application. Indicate directly on the boxplot which groups are significantly different from each other based on above R output. Use the letter scheme shown in lecture. [2 marks] E. Use the medians in the above boxplot to draw the interaction plot below. Draw the figure to scale and be as accurate as possible. [2 marks] D.Cream E.Cream D.Gel E.Gel D.Sol E.Sol 0 10 20 30 40 Color Change a b a b c b 10 15 20 25 30 Formulation Color Change Dihhydroxyacetone Erythrulose Application Solution Gel Cream ADD BLANK BOX Downloaded by Kirsten Schneider (k.schneider4696@gmail.com) lOMoARcPSD|13275606

13 25) A study was published in the Lancet on the body mass index (BMI) of people from around the globe. The following table shows a random subset of the data for 10 people. For this question, test the hypothesis that the mean BMI of people in this sample is above the ‘healthy’ threshold of 25. 25.76 25.72 25.39 24.65 24.90 26.01 25.97 25.11 25.91 26.54 A. Indicate the statistical test that is most appropriate for this data and the scientific question [1 mark] [single sample t-test] B. Write the null and alternative hypotheses [1 mark] [Ho: m <= 25, Ha: m > 25, where m is the sample BMI mean. The sign is important here] C. Calculate the observed test score. Make sure to show your work and report your answer to three decimal places. [3 marks] [m=25.596, sd=0.578, n=10, t=(m-25)/(sd/sqrt(n)), t=3.264] D. If the critical score is 1.833, write your statistical conclusions and your scientific conclusions. Include your rationale for the statistical conclusions. [2 marks] [Since the t-observed is greater than t-crit=1.833, we reject the null hypothesis. We conclude that there is evidence to support that the mean BMI in the sample is greater than 25.] E. In the space below, draw the null distribution and null hypothesis for the statistical test along with proper axis labels. Label both the null distribution and the null hypothesis. Draw on the observed test score and shade in the region that represents the p-value. [3 marks] Remove picture Downloaded by Kirsten Schneider (k.schneider4696@gmail.com) lOMoARcPSD|13275606

14 26) For each of the following studies, identify and rationalize the most appropriate statistical test (e.g., t-test, regression etc.). Include the null and alternative hypotheses (be mindful of direction in the test), as well as the appropriate test statistic (e.g., F- score for an F-test). STUDY 1 A geneticist was interested in the degree to which behavior is determined by gender. She conducted a survey of 40 random students. The scientist categorized each student as being male or female, and their study behavior as diligent, procrastination or reactive. She wants to know whether study style depends on gender i. Identify and rationalize the most appropriate statistical test [1 mark] This is a Chi-square analysis because both factors are categorical. ii. State the null and alternative hypothesis [1 mark] Ho: there is no difference between expected and observed frequencies. HA: there is a difference between expected and observed frequencies. (students could also use the independent/non independent terminology) iii. Identify the appropriate test statistic [1 mark] This is a Chi-square test and the test statistic is the Chi-square score. STUDY 2 A sports medicine study was conducted looking at whether changing the source of dietary proteins improved race times in marathon runners. The study asked 50 racers to increase the proportion of their dietary protein coming from plants for a season. The researchers recorded the proportion of dietary protein derived from plants (proportion) and the change in race time (minutes) for each racer. They wanted to test the hypothesis that increasing the proportion of plant proteins could predict faster race times. i. Identify and rationalize the most appropriate statistical test [1 mark] This is a linear regression because both variables are numerical and the question is about prediction. ii. State the null and alternative hypothesis [1 mark] The null hypothesis is H0: b ≥ 0, HA: b<0, where b is the slope. iii. Identify the appropriate test statistic [1 mark] The test is a t-test, and the test statistic is a t-score. Downloaded by Kirsten Schneider (k.schneider4696@gmail.com) lOMoARcPSD|13275606

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15 26) continued… STUDY 3 Researchers are interested in whether a new form of bio control can successfully manage pest insects on tea plantations compared to using pesticides. They conduct an experimental study where 120 tea plants are randomly allocated to either i) no biocontrol or pesticide, ii) biocontrol but no pesticide, and iii) pesticide but not biocontrol. For each tea plant, they measured the mass of consumed after exposure to the insects. i. Identify and rationalize the most appropriate statistical test [1 mark] This is single-factor ANOVA because there is one categorical factor and a numerical response variable. ii. State the null and alternative hypothesis for the question whether there is an effect of biocontrol. [1 mark] The null hypothesis is H0: u1=u2=u3, HA: u1 ≠ u2 ≠ u3, where u is the mean for each treatment. iii. Identify the appropriate test statistic [1 mark] The test statistic is an F-score. 27) Maple syrup production is looking to be good this year for sugar bush operators. You are interested in how soil type impacts the amount of maple syrup produced per hectare of forest, so conduct a study where you randomly sample 60 maple sugar bush operators from each municipality on Ontario and Quebec. For each operator, you record the soil type (humic, acidic, sandy, bog) and the amount of production (<100L per ha, 100-200L per ha, >200L per ha). A. Indicate the survey design used in the study [1 mark] [stratified sampling] B. Indicate the sampling unit and observation unit [1 mark] [0.5 marks for each of sampling unit: sugar bush, observation unit: sugar bush] C. Indicate the type of data for each of the measurement variables in the study [1 mark] [0.5 marks for each of soil type: categorical, production: categorical] Downloaded by Kirsten Schneider (k.schneider4696@gmail.com) lOMoARcPSD|13275606

16 27) continued… D. The following table shows the data from the Maple Syrup Study for one of the municipalities in question 27. Humic soils Acidic soils Sandy soils Bog soils <100L per ha 8 12 5 4 100-200L per ha 13 5 1 1 >200L per ha 5 3 1 2 i. Indicate the statistical test that is most appropriate for this data and the scientific question [1 mark] [Chi-square test] ii. Write the null and alternative hypotheses [1 mark] [Ho: soil type is independent of maple syrup production, Ha: soil type is not independent of maple syrup production. iii. Calculate the missing expected counts in the table below under the null hypothesis. Report your answers to one decimal place, showing your work in the space below the table for full marks. [3 marks] [For the first cell, it would be ((8+13+5)/60)*((8+12+5+4)/60)*60=12.6. Full table is shown below] Expected Humic soils Acidic soils Sandy soils Bog soils <100L per ha 12.6 9.7 3.4 3.4 100-200L per ha 8.7 6.7 2.3 2.3 >200L per ha 4.8 3.7 1.3 1.3 Downloaded by Kirsten Schneider (k.schneider4696@gmail.com) lOMoARcPSD|13275606

17 27) continued… iv. Calculate your observed test score to two decimal places. [2 marks] [Observed test score is 7.70] df=(number of rows -1)*(number of columns - 1)=6] v. Find the critical score and write both your statistical conclusions and scientific conclusions. [2 marks] [chisq-crit=12.592. Since the chisq-observed is less than chisq-crit, we fail to reject the null hypothesis. We conclude that there is no evidence that maple syrup production and soil type are not independent.] 28) Explain what a null F-distribution represents. Include a clear definition of the F-score (i.e., how is an observed F-score calculated), and then include a clear description of what the null distribution for the F-score represents. Be as specific as possible. [3 marks] [The F-score is the ratio of the variation among categorical groups divided by the residual variation within a group. The null distribution for the F-score represents the variation in that ratio you would expect from repeated sampling of a population where there was no true difference in the means.] Downloaded by Kirsten Schneider (k.schneider4696@gmail.com) lOMoARcPSD|13275606

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Final Exam practice (4)

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