FInals Quanti Review

xlsx

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Confederation College *

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MISC

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Statistics

Date

Jan 9, 2024

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xlsx

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36

Uploaded by BarristerWillpowerKoala27

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Match each scatterplot shown below with one of the four specified correlations.
A regression was run to determine if there is a relationship between hours of TV watched per day (x) and num The results of the regression were: Given: ˆy= bx+a b= -1.244 (slope) a= 20.31 (y-intercept) R2= 0.471969 (R-squared) a) Use this to predict the number of situps a person who watches 5.5 hours of TV can do (to 3 decimal plac x= 5.5 number of hours of TV watched per day (independent variable) Answer: 13.468 predicted number of situps a person can do who watches 5.5 hour of TV b) What is the value of the correlation coefficient (to 3 decimal places) ? Answer= -0.687 correlation coeeficient
mber of situps a person can do (y). ces)
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Phone Check-Ins (x) 34 42 58 Quiz Grade (y) 4.4 5.5 3.8 Phone Check-ins (x) Quiz Grades (y) COUNT (number of observation)= 8.0000 df= 6.0000 α= 0.0500 r= -0.9083 b1 (slope)= -0.0544 bo (intercept)= 6.9049 a) What is the correlation coefficient? -0.9083 b) Perform a hypothesis test to determine whether the correlation is statistically significance. -5.3175 2.447 *answer should be 2.4470 since rounde Yes c) Find the equation for the regression line. Keep at least 4 decimals for each parameter in the equation. slope= -0.0544 Anderson believes that students are distracted by their phones and hence students are not performing as w app on their phone that tracks the number of times they look at their phone per day. Test to see if there is a a student checks their phone and a student's quiz score using α = 0.05. The student data is displayed in the between steps. Round answers to 4 decimal places . r = Test Statistic: Critical Value: Is the correlation significant?
intercept= 6.9049 ˆy= -0.0544x + 6.9049 (this is the equation) -0.0544x+6.9099 *this is the answer d) Interpret the slope in context. Answer: Anderson should expect a grade change of -0.0544 for each additi e) What does the equation predict as the grade for someone with 42 phone check-ins? x = 42 number of phone check-ins Grade= 4.6182 Actual grade = 5.5 Residual Error= 0.8818 (A slope of -0.0544 means that for every one unit increase in phon decrease of 0.0544 points in the predicted quiz grade. This indicat phone check-ins and quiz grades in your data. f) One observation in the data set was a student who checked their phone 42 times and scored 5.5 on the the residual error for the point (42,5.5)?
59 76 80 83 83 3.4 2.8 3 2.2 2.1 ed off to always 3 decimals well in class. Students are asked to install an a relationship between the number of times table and graph below. Do not round
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tional phone check-in. ne check-ins, there is an average tes a negative correlation between e quiz. What is
Excel Output: Regression Statstics Significance level 0.01 Multiple R 0.762 R Square 0.5806 Adjusted R Square 0.5764 Standard Error 1.681 Observations 100 ANOVA df SS MS F Significance F Regression 4 371.6747 92.9187 32.8837 0 Residual 95 268.4392 2.8257 Total 99 640.1139 Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Intercept 51.5585 5.3176 9.6958 0 37.5808 65.5362 Weight 0.0417 0.0337 1.2387 0.21852 -0.0468 0.1303 Wrist Circ. 1.3678 1.7179 0.7962 0.42791 -3.1479 5.8835 Gender -0.0244 1.2198 -0.02 0.9841 -3.2307 3.182 The first three questions are about the wrist circumference variable. Regression Coefficient of the wrist circumference= 1.3678 p-value of the wrist circumference= 0.4279 Answer: No No, wrist circumference is not statistically significant at a 0.01 level The remaining questions are about the model as a whole. Percentage of the Variation in Heaight = 41.94% A research team believes that height (in inches) can be predicted using weight (in pounds), wrist circumference and gender (a categorical variable that is 0 for females and 1 for males). They took measurements for 100 individ selected females and 50 randomly selected males) and performed a multiple linear regression analysis using Exce displayed below. a) What is the regression coefficient of the wrist circumference variable? Express your answer with four decima b) What is the p-value of the individual coefficient t test for the wrist circumference variable? Express your answer as a decimal rounded to four decimal places. c) Based on the p-value, is this variable significant at a 0.01 significance level? d) What percentage of the variation in height is left unexplained by this model? Express your answer as a perce * The R-squared value is 0.5806, meanin the model. Therefore, 100%−58.06%=41 e) One individual in the dataset has a weight of 161.19 lbs., a wrist circumference of 6 in., and is female. Using regression model from the output, what would the predicted height be? Express your answer rounded to the n hundredth of an inch (i.e. two decimal places).
Weight= 161.19 * use the regression equation Wrist Circumfernce= 6 Gender 0 66.49 inches Predicted height for given individual Height= 66.8 0.31 inches Residual error for the individual Answer: Weight 0.21852 Most Statistically Significant Variable for Predicting * manual comparison or checking Answer: Gender 0.9841 Least Statistically Significant Variable for Predicting * manual comparison or checking Answer: Yes, it should be removed j) With gender being a categorical variable, how would you interpret the gender coefficient? Answer: * gender (a categorical variable that is 0 for females and 1 for males ˆy= f) The same individual from part e) had an actual height of 66.8 inches. What is the residual error for this data p Express your answer rounded to the nearest hundredth of an inch (i.e. two decimal places). y−ˆy= g) Which variable is the most statistically significant for predicting height? h) Which variable is the least statistically significant for predicting height? i) Should this least statistically significant variable be removed from the model based on a 0.01 significance leve The gender coefficient of -0.0244 indicates that, all else being equal, males (coded as 1) inches shorter in height than females (coded as 0). However, given the high p-value asso difference is not statistically significant at conventional levels, suggesting caution in inte gender effect on height within this model.
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of significance (in inches), duals (50 randomly el. The output is al places. entage rounded to two decimal places. ng 58.06% of the variation in height is explained by 1.94%100%−58.06%=41.94%. the nearest
Height (* The variable with the smallest p-value is the most statistically significant.) Height (* The variable with the largest p-value is the most statistically significant) point? el? are predicted to be, on average, 0.0244 ociated with this coefficient, this erpreting this result as evidence of a true
Significance level: 0.05 41 11 52 df= 2 20 7 27 30 26 56 91 44 135 a. Choose the correct null and alternative hypotheses. Answer: b. Compute the test statistic. 35.0519 16.9481 18.2000 8.8000 37.7481 18.2519 1.0094 2.0876 0.1780 0.3682 1.5904 3.2892 Compute the value of the test statistic. (Round your answer to 4 decimal places.) χ2stat= 8.5227 c. Compute the critical value. (Round your answer to 3 decimal places.) χ2crit= 5.991 Interpret the results of the significance test Answer: (* use IF statement and put critical value and then stat value) The color of the Star Trek uniform represents each crew-member's work area. In the original Star Trek series, blue shirts are worn by medical and science staff, gold shirts are worn by the command division, and the red shirts were worn by engineering, security and communications division. We will statistically assess whether there is a connection between uniform color and the fatality rate. The table below shows a sample of how many crew-members in each area have died onscreen and their shirt color. Use α = 0.05 to test to see if a crew-member's fatality onscreen is dependent on their uniform color. Alive Dead Total Blue Gold Red Total H 0: There is no association between Star Trek fatalities and uniform color. Ha : There is an association between Star Trek fatalities and uniform color. Complete the following table of expected frequencies . (Round your answers to 4 decimal places). Alive Dead Blue Gold Red Complete the following table of Pearson residuals . (Round your answers to 4 decimal places). Alive Dead Blue Gold Red The test statistic provides strong evidence against the null hypothesis. The association between Star Trek fatalities and uniform color is statistically significant.
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Observed Expected fo fe (fofe)^2 fe Significance level: 0.01 * 1% as stated above 1-7 43 43 0.0000 8-14 43 43 0.0000 15-21 42 43 0.0233 22-28 42 43 0.0233 29-35 45 43 0.0930 0.1395 4 13.277 Decision: Fail to Reject the Null Hypothesis Conclusion: There is not enough evidence *A The results for the Powerball are not equally likely *B Powerball is a large lottery in which players try to guess the numbers that will turn up in a drawing of numbered balls. One of the balls drawn is the "Powerball". If you select the correct number on the Powerball, your winnings will be increased. The possible numbers for the Powerball are 1 through 35. If the selection is random, then there should be an equal number of draws for the Powerball across the following 5 categories: 1-7, 8-14, 15-21, 22-28, 29-35. Using a 1% significance level, test that Powerball numbers are equally likely. Round to four decimal places unless otherwise specified. Powerball Number Pearson Residuals Test Statistic: Degrees of Freedom: Critical Value:
Significance level: 0.05 Blood Type Percentage Observed Expected fo fe (fofe)^2 fe A 41% 110 120.950 0.991 B 10.90% 39 32.155 1.457 AB 2.20% 6 6.490 0.037 O 45.90% 140 135.405 0.156 Total 295 χ2stat= 2.641 * x2stat is from pearson residuals df= 3.000 χ2crit= 7.815 d. What decision should be made? Decision: The test statistic is less than the critical value. The researcher should fail to reject the null hy e. What should the final conclusion be? Conclusion: In Canada, 41% of all people have type A blood, 10.9% have type B blood, 2.2% have type AB blood and 4 have type O blood. A researcher wants to see if the blood type distribution is different for millionaires. Th below shows the results of a random sample of 295 millionaires. What can be concluded at an α = 0.05 sig level using the following hypotheses? H0: The distributions of blood types are the same between the general population and millionaries. H1: The distributions of blood types are NOT the same between the general population and millionaries. a. You sampled 295 millionaires about their blood type; the observed frequencies are recorded below. C the table below by filling in the expected frequencies and the Pearson residuals. Express your answers r to three decimal places . Pearson Residuals b. Calculate the test-statistic for this data. Express your answers rounded to three decimal places . c. What are the degrees of freedom and the critical value for this test? Express your answers rounded to Based on the sample data, there is not sufficient evidence to conclude that the distributions types are not the same between the general population and millionaires at an α = 0.05 signifi level
ypothesis 45.9% he table gnificance Complete rounded o three decimal places if needed. s of blood ficance
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This test is using the hypotheses: Significance level: 0.05 Sample size (n): 20 H0:p≥0.31 No. of successes (O+ donrs) 80 H1:p<0.31 Hypothesized Portion: 31% a) Is this a one-tailed or a two-tailed test? Answer: a one-tailed test zcrit: -1.645 4 Test statistic: 35.681 d) Based on the above, what would you conclude? Decision: O+ blood is given to patients more than any other blood type, making it the most needed blood type. Approximately 31% of the population has O+ blood. Medical researchers are interested in testing whether the number of O+ blood donors is proportional to the 31% of people who have O+ blood type. One particular day, a local blood donation clinic takes donations from 80 donors, of which 20 were donors with O+ blood. Use a binomial test and a significance level of α=0.05 to determine whether the proportion of donors with O+ blood is lower than the proportion of people with O+ blood in the population. * This is a one-tailed test because the alternative hypothesis ( H 1: p <0.31) sp b) What is the critical value, zcrit , found in the z- or t-table? * the hypothesis is testing whether the proportion of O+ blood donors is less than t This means we are interested in the lower tail of the normal distribution , as we are c) What is the value of the test statistic for this scenario? Express your answer rounded to three decimal places . Sample Proportion ( p) : Fail to reject the null hypothesis since the test statistic is greater than or equal to the critical value Therefore, I cannot conclude that the proportion of O+ donors is lower than the proportion of peo O+ blood type in the population
pecifies a direction – it's testing if the proportion is less than 0.31. the proportion in the general population (31%). e looking for a significantly lower proportion than the hypothesized 31%. e. ople with
Significance level: 0.05 Sample size (n): 70 No. of successes (red wins44 50% a) What are the appropriate hypotheses for this test? Answer: b) What is the critical value found in the z- or t-table? zcrit= 1.645 c) What is the value of the test statistic for this scenario? 0.6286 zstat= 2.151 d) Based on the above, what would you conclude? Red is a colour viewed as dominating and powerful. Researchers were interested in whether athletes wearing red were more likely to win than athletes not wearing red. The outcome of four combat sports (boxing, Tae Kwon Do, Greco-Roman wrestling, and freestyle wrestling) were monitored during the 2004 Olympic games and it was found that the combatant wearing red won more often than the combatant wearing blue: 44 out of 70 observed matches. Use a binomial test to determine whether this is sufficient evidence to prove that athletes wearing red win significantly more often than athletes wearing blue. Use α=0.05 and round your answers to three decimal places as needed. Let p represent the proportion of matches won by the combatant wearing red. *match es Hypothesized Portion ( po ) H0: p ≤0. wearing red, so the probability of winning is equal to 0.5 (50%). The alternative hypothesis (H1) tests if wearing red gives a higher probability of winning than 50%. H1: p >0. Sample Proportion ( p) : Reject the null hypothesis since the test statistic is greater than the critical value. Thus, I conclude that there is sufficient evidence that the combatant wearing red wins more frequently
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Excel Output: Regression Statstics Multiple R 0.8663 R Square 0.7504 Adjusted R Square 0.7365 Standard Error 63.4792 Observations 20 ANOVA df SS MS F Regression 4 181730.8176 45432.7044 11.2747 Residual 15 60444.1824 4029.6122 Total 19 242175 Coefficients Standard Error t Stat P-value Intercept 1211.2421 55.5906 21.7886 0 Bedrooms 76.1635 13.4965 5.6432 4.70E-05 Bathroom -15.2201 13.8393 -0.478 0.639521 Utilities 109.717 29.6496 3.7004 0.002137 a) Write out the regression equation using x1, x2, and x3 as the variables. Do not round the coefficien ˆy= 1211.2421 + 76.1635 * x1-15.2201*x2+109.717*x3 b) Interpret each of the four coefficients within the context of the question: Intercept (1211.2421): This is the base rent when the number of bedrooms, bathrooms, and utilities inc Bedrooms (76.1635): For each additional bedroom, the rent is expected to increase by $76.1635, holdin Bathrooms (-15.2201): For each additional bathroom, the rent is expected to decrease by $15.2201, ho Utilities (109.717): If utilities are included (x3 = 1), the rent is expected to increase by $109.717, compa Percentage of the Variation: 75.04 R Square Expcted Rent: 1443 bedrooms The most statistically significant variable is bedrooms, as it has the lo A statistics student is looking for an apartment to rent in Malton. Knowing what she does about regress online listings and construct a regression model to get a sense of the rental market. The number of bed not utilities are included in the rent (0 = not included, 1 = included) are what she believes are the most The student reviews 20 rental listings in the Malton area, records the key information, and uses Excel to displayed below. c) What percentage of the variation in rent is explained by this model? Express your answer as a perc d) Using the regression equation, what would you expect the rent to be for a 2 bedroom, 2 bathroom e) According to the regression output, which variable is the most statistically significant variable for p f) According to the regression output, which variable is the least statistically significant variable for pr
bathrooms The least statistically significant variable is bathrooms, as it has the h The next four questions are about the bedrooms variable. Coefficient of Bedrooms: 76.1635 P-value of Bedroom: 4.70E-05 rounded to 4 will be 0.0000 i) Based on the p-value, is the bedrooms variable significant at a 0.01 significance level? Yes the bedrooms variable is statistically significant at a 0.01 level of sign j) Should this variable be removed from the model? No bedrooms should not be removed from the model. The variable is sta g) What is the regression coefficient of the bedrooms variable? Express your answer with four decima h) What is the p-value of the individual coefficient t test for the bedrooms variable? Express your ans
Significance F 0.0002 Lower 95% Upper 95% 1047.4325 1375.0518 36.3931 115.9339 -109.0413 78.6011 22.348 197.086 nts. cluded are all zero. ng all other factors constant. olding all other factors constant. This might need further investigation on what other variables affecting re ared to when utilities are not included, holding all other factors constant. owest P-value (4.7E-5). sion, she decides to collect information from the drooms, the number of bathrooms, and whether or important factors for determining the rent price. o conduct a regression analysis. The output is cent with two decimal places . m apartment if utilities are included? Round your answer to the nearest whole number . predicting rent? redicting rent?
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highest P-value (0.639521). nificance, since its P-value (4.7E-5) is less than 0.01. atistically significant as indicated by its low P-value, suggesting it plays an important role in predicting re al places. swer as a decimal rounded to four decimal places.
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ent prices.
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ent.
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α= 0.05 a) What are the appropriate hypotheses for this test? H0:p≤0.2 H1:p>0.2 b) What is the critical value found in the z- or t-table? zcrit 1.645 1.64485363 c) What is the value of the test statistic for this scenario? 0.314286 zstat= 2.390 d) Based on the above, what would your decision and conclusion be? Decision: Reject the null hypothesis Conclusion: there is not enough statistical evidence to suggest Hayden's performance wa Hayden is taking a first-year university psychology course with a final exam that consists of 70 multiple choic question has five possible answers, only one of which is correct. If Hayden scored 22 out of 70, did he do be expected by chance (i.e. guessing at each question)? Use a one-tailed binomial test with α=0.05 to determine this. Round all answers to three decimal places wh represent the proportion of questions that Hayden answered correctly. p ^ =
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as better than chance. ce questions. Each etter than would be hen needed. Let p
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Significance level: 0.05 Dwelling Type Percentage Observed Expected fo fe (fofe)^2 fe Row/Townhouse 50% 81 86.688 0.373 Semi-detached 34.50% 71 59.340 2.291 Detached 15.10% 20 25.972 1.373 Total 172 χ2stat= 4.038 * x2stat is from pearson residuals df= 2.000 χ2crit= 5.991 d. What decision should be made? Decision: The test statistic is less than the critical value. The researcher should fail to reject the null hy Last year in Manitoba, 50.4% of first-time home buyers purchased row/townhouse-type homes, 34.5% of first-tim buyers purchased semi-detatched-type homes, and 15.1% of first-time homebuyers purchased detached-type ho an attempt to empower first-time home buyers, the provincial government implemented a number of new polici whether these policy changes are having an impact, a random sample of first-time home buyers was taken from a time home buyers since the policy changes were implemented. The purchased dwelling type was recorded for th time home buyers. The results are shown in part b. below. Can we conclude that the policy changes implemented by the Manitoba government have changed the distributi types of dwellings purchased by new home buyers? Conduct a chi-square test using α=0.05 to determine this. H0: The distributions of blood types are the same between the general population and millionaries. H1: The distributions of blood types are NOT the same between the general population and millionaries. a. Which pair of hypotheses would be used for this test? rounded to three decimal places . H 0: The proportions have not changed since the new policies were implemented. H 1: The proportions have changed since the new policies were implemented. b. A random sample of 172 first-time home buyers (since the policy change) was taken and the dwelling type p was recorded. The observed frequencies are summarized below. Complete the rest of the table by filling in the expected frequencies and the Pearson residuals. Express your an rounded to three decimal places . Pearson Residuals b. Calculate the test-statistic for this data. Express your answers rounded to three decimal places . c. What are the degrees of freedom and the critical value for this test? Express your answers rounded to three
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e. What should the final conclusion be? Conclusion: Based on the sample data, there is not enough sufficient evidence to suggest that the policy implemented by the Manitoba government have changed the distribution of the types of dw purchased by new home buyers
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ypothesis me home omes. In ies. To see all first- hese first- tion of the purchased nswers decimal places if needed.
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y changes wellings
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Day Monday Tuesday Wednesday Phone Check-Ins (x) 29.5 28.5 29.1 Quiz Grade (y) 2187.56 2163.55 2064.34 Phone Check-ins (x) Quiz Grades (y) COUNT (number of observation)= 5.0000 df= 3.0000 α= 0.0500 r= 0.9785 SSx= 32.7080 Ssy= 283928.3139 SSp= 2981.7976 Correlation Coefficient Interpretation 0.9785 The correlation coefficient of 0.9785 indicates a very strong positive lin A beachside ice cream shop is investigating the effect weather has on its ice cream sales. Recognizing that weekd sales, the owner decides to track the midday temperature (in °C) from Monday to Friday over the course of one The findings are summarized in the table and graph below. a) Calculate and interpret the correlation coefficient by first calculating SS X , SS Y , and SP . Round your answers to four decimal places . r = b) Perform a hypothesis test to determine whether the correlation is statistically significant. Use a of α = 0.1. Round both answers to three decimal places .
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8.211 2.353 Yes c) Calculate the equation for the regression line. Round each coefficient to four decimal places and use these r slope (b)= 91.1642 intercept (a)= -513.3060 ˆy= -513.306 + 91.1642x (this is the equation) d) Interpret the slope in context. Answer: The slope of 91.1642 suggests that for each additional degree Celsius e) What does the equation predict the sales will be for a day with a midday temperature of 28.5°C? Use the ro x = 28.5 Prediction 2084.87 Residual Error= 78.68 Test Statistic: Critical Value: Is the correlation significant? f. One of the five days observed had a midday temperature of 28.5°C and sales of $2163.55. What is the residu (28.5,2163.55)? Round your answer to two decimal places.
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Thursday Friday 23 30 1575.46 2214.66 near relationship between temperature and daily sales. days and weekends could have very different week and matches this to the daily sales. a significance level
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rounded values in the equation. Hint: Some of the values from above may be useful. in temperature, the daily sales increase by approximately $91.16. ounded coefficients for this calculation and round your answer to two decimal places. ual error for the point
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Punched Card Type Redeemed Not Redeemed Redeemed Observed Expe fo f Buy-8-Get-1 Free 10 40 50 15.5000 Buy-10-Get 1 Free 21 29 50 15.5000 Total 31 69 100 Test Statistic = 5.6568 Critical Value = 3.841 Decision: Reject H0 Conclusion: A local coffee shop is trying to promote repeat business by launching a buy-eight-get-one-free rewards coffee after eight purchases. They are considering a variation of this promotion that may encourage cu free card with two free punches. Both provide a free coffee after eight purchases, but the second versi initial test, the coffee shop randomly distributed 50 of each card to customers and tracked how many o months. Use the data below, along with a significance level of α = 0.05, to test whether the number of redeeme punch card issued. Total for Card Types There is evidence to suggest that there is an association between the redeemed cards. This implies that the type of punch card issued does
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Not Redeemed Pearson Residuals ected Redeemed Not Redeemed fe (fofe)^2 (fofe)^2 fe fe 34.5000 1.9516 0.8768 34.5000 1.9516 0.8768 s punch card that provides customers with a free ustomers to come back more: a buy-ten-get-one- ion gives the appearance of more progress. In an of each punch card were redeemed after three ed punch cards is dependent on the type of e type of punch card and the number of s affect the likelihood of it being redeemed.
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