Business Statistics (BK/IBA) Tutorial 4 Full solutions

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1 Business Statistics (BK/IBA) Tutorial 4 Full solutions Instruction In a tutorial session of 2 hours, we will obviously not be able to discuss all questions. Therefore, the following procedure applies: we expect students to prepare all exercises in advance; we will discuss only a selection of exercises; exercises that were not discussed during class are nevertheless part of the course; students can indicate their wish list of exercises to be discussed during the session; teachers may invite students to answer questions, orally or on the blackboard. We further understand that your time is limited, and in particular that your time between lecture and tutorial may be limited. In case you have no time to prepare everything, we kindly advise you to give priority to the exercises that are indicated with the are not relevant! 7A+B Several μs: comparison + Several μs and medians: more issues icon. This does not mean that the other questions Q1 (based on Doane & Seward, 4/E, 11.7 and 11.13) Semester GPAs are compared for seven randomly chosen students in each class level at Oxnard University. a. Does the data show a significant difference in mean GPAs? b. Which pairs of mean GPAs differ significantly (4 majors)? Excel output for ANOVA Anova: Single Factor SUMMARY Groups Count Sum Average Variance Accounting 7 19,84 2, , Finance 7 21,17 3, , Human Resources 7 22,69 3, , Marketing 7 23,6 3, , ANOVA Source of Variation SS df MS F P-value F crit Between Groups 1, , , , , Within Groups 2, , Total 3, SPSS output for post-hoc BS 1 Tutorial 4

2 Q1 a. Yes; b. the mean score for Marketing differs significantly from the mean score for Accounting. Sol a. Please use 5 steps procedure and use α = step 0 Model: Y ij = μ + α i + ε ij with ε~n(0, σ 2 ) (i) The hypotheses to be tested are: H 0 : α 1 = α 2 = α 3 = α 4 = 0 versus H 1 : not H 0 (α = 0.05) (ii).sample Statistic: F = MSA ; reject for large values MSW (iii) Distribution test statistic under H 0 : F~F 3,24 Requirement: all four populations normal; all four variances equal (which we will assume). (iv) Calculated test statistic: F calc = Reported p-value: ; p-value of this statistical problem: Critical value: F crit;upper = F 3,24;0.05 = 3.01 (from table) or (from Excel) (v) We reject the null hypothesis since the test statistic exceeds the critical value. This is a fairly close decision. The p-value of is less than The difference between the GPA means most likely did not occur by chance. From the dot plot (below), we see the GPA for Accounting below the overall mean and Human Resources and Marketing above the overall mean. BS 2 Tutorial 4

3 Extra Q2 b. At α = 0.05, the mean score for Marketing differs significantly from the mean score for Accounting (in fact: it is higher). All other pairs do not differ significantly. In a five-step procedure at the exam, you should use the critical value approach or the p-value approach. It is not needed to use both. In our solutions, we sometimes write down both approaches, to help you check your solution. The retailing manager of a supermarket chain wants to determine whether product location has any effect on the sale of pet toys. Three different aisle locations are considered: front, middle, and rear. A random sample of 18 stores is selected, with 6 stores randomly assigned to each aisle location. The size of the display area and price of the product are constant for all stores. At the end of a one-month trial period, the sales volumes (in thousands of dollars) of the product in each store were as follows (and are stored in the file Locate). Aisle Location Front Middle Rear 8,6 3,2 4,6 7,2 2,4 6,0 5,4 2,0 4,0 6,2 1,4 2,8 5,0 1,8 2,2 4,0 1,6 2,8 a. At the 0.05 level of significance, is there evidence of a significant difference in mean sales among the various aisle locations? b. If appropriate, which aisle locations appear to differ significantly in mean sales? c. At the 0.05 level of significance, is there evidence of a significant difference in the variation in sales among the various aisle locations? d. What should the retailing manager conclude? Fully describe the retailing manager s options with respect to aisle locations. Test of Homogeneity of Variances SALES Levene Statistic df1 df2 Sig. 2, ,142 BS 3 Tutorial 4

4 ANOVA SALES Between Groups Within Groups Total Sum of Mean Squares df Square F Sig. 48, ,222 14,105,000 25, ,717 74, Dependent Variable: sales Multiple Comparisons Tukey HSD (I) locate front middle rear (J) locate middle rear front rear front middle *. The mean difference is significant at the.05 level. Mean Difference (I-J) Std. Error Sig. 4,0000*,7566,0003 2,3333*,7566,0195-4,0000*,7566,0003-1,6667,7566,1031-2,3333*,7566,0195 1,6667,7566,1031 sales Subset for alpha =.05 locate N 1 2 Tukey HSD a middle 6 2,067 rear 6 3,733 front 6 6,067 Sig.,103 1,000 Means for groups in homogeneous subsets are displayed. a. Uses Harmonic Mean Sample Size = 6,000. Sol Q2 a. Yes; b. μfront differs significantly from μmiddle and μrear; c. No; d. The front aisle is best. a. Please use 5 steps procedure. Model: Y ij = μ + α i + ε ij with ε ij ~N(0, σ 2 ) To test at the 0.05 level of significance whether the average sales volumes in thousands of dollars are different across the three store aisle locations, we conduct an F test (ANOVA). (i) H 0 : α front = α middle = α rear = 0 versus H 1 : not H 0 (α = 0.05) (ii) Sample statistic: F = MSA ; reject for large values MSW (iii) Distribution (standardized) test statistic under H 0 : F~F 2,15 Required: all three populations normal; all three variances equal. To test normality: no computer output available; to test equal variances: see output Test of Homogeneity of Variance, which suggests the variances are equal (p-value of 0.142). (iv) Calculated test statistic: F calc = Critical value: F upper = F 2,15 = 3.68 Reported p-value: 0.000, p-value of this statistical problem: BS 4 Tutorial 4

5 Extra (v) Decision: Since F calc = is above the critical bound of F crit = 3.68, reject H 0. There is enough evidence to conclude that the average sales volumes in thousands of dollars are different across the three store aisle locations. b. The three means are: y middle = 2.067, y rear = and y front = To determine which of the means are significantly different from one another, we use the Tukey-Kramer procedure. SPSS gives the homogeneous subsets with α = 0.05: μ front differs significantly from μ middle and μ rear, while μ middle and μ rear do not differ significantly from each other = σ rear = σ front versus H 1 : At least one variance is different (α = 0.05) c. (i) H 0 : σ middle (ii) Use Levene s F-ratio for homogeneity of variance. (iii) Under H 0 we have F~??? (SPSS computations). We do not need further assumptions on the distributions of all three populations. Specifically, in contrast to the usual two-sample Ftest for variances, normal populations are not necessary. (iv) Use the SPSS output for Levene s test for homogeneity of variance: F calc = and p-value = P(F 2.227) = (v) Since the p-value = > 0.05, do not reject H 0. There is no evidence of a significant difference in the variation in sales among the various aisle locations. This suggests that the use of the ANOVA is justified, at least it does not lead to the conclusion that it is not justified. d. The front aisle is best for the sale of this product. The manager should evaluate the trade-off in switching the location of this product and the product that is currently intended for the front location. In a five-step procedure at the exam, you should use the critical value approach or the p-value approach. It is not needed to use both. In our solutions, we sometimes write down both approaches, to help you check your solution. Q3 (based on Doane & Seward, 4/E, 16.10) The results shown below are mean productivity measurements (average number of assemblies completed per hour) for a random sample of workers at each of three work stations. a. At α =.05, is there a difference in median productivity? b. Use one-factor ANOVA to compare the means. c. Do you reach the same conclusion? BS 5 Tutorial 4

6 Sol Q3 a. reject H0; b. reject H0; c. the same. a. Use the Kruskal-Wallis non-parametric ANOVA. (i) H 0 : M A = M B = M C ; H 1 : not H 0 ; α = 0.05 (ii) Sample statistic: use mean rank sums T A, T B, and T C to compute H-statistic; reject for large values 2 (iii) Under H 0 : H~χ 2 Requirement: n for all groups 5: OK (iv) χ calc = 9.479; χ crit = χ upper;0.05;2 = Reported p-value: (v) Reject H 0 because χ calc χ crit (or because p-value α) b. Use the parametric ANOVA. (i) H 0 : μ A = μ B = μ C ; H 1 : not H 0 ; α = 0.05 (ii) Sample statistic: F = MSA ; reject for large values MSW (iii) Under H 0 : F~F 2,22 Requirement: normal populations, all with equal variances (which we will assume) (iv) F calc = 7.718; F crit = F upper;2,22;0.05 = 3.44 Reported p-value: BS 6 Tutorial 4

7 Extra (v) Reject H 0 because F calc F crit (or because p-value α) c. The final conclusion is the same. Both tests are quite convincing. The Kruskal-Wallis is slightly more conservative. In a five-step procedure at the exam, you should use the critical value approach or the p-value approach. It is not needed to use both. In our solutions, we sometimes write down both approaches, to help you check your solution. Q4 (based on Berenson 11/E, 11.38) A student team in a business statistics course performed an experiment to investigate the time required for pain-relief tablets to dissolve in a glass of water. The factor of interest is the temperature of the water (hot or cold). The experiment consisted of 12 replicates for each of the two temperature levels. The following data show the time a tablet took to dissolve (in seconds) for the 24 tablets used in the experiment: At the 0.05 level of significance, a. Is there an effect due to water temperature? Use Output from SPSS: BS 7 Tutorial 4

8 b. Can you redo this analyses with a t-test? Q4 a. there is a temperature effect ; b. the analysis can be repeated with an independent samples t-test. Sol a. Use SPSS output, with model Y ij = μ + α i + ε ij and ε ij ~N(0, σ 2 ) (i) H 0 : α H = α C = 0; H 1 : not H 0 ; α = 0.05 (ii) Sample statistic: F = MSA ; reject for large values MSW (iii) Under H 0 : F~F 1,22 ; requirements: normal populations with equal variances (which we will assume) (iv) F calc = ; F crit = 4.30; p-value = (Mind that F calc is not to scale) (v) reject H 0 without any doubt: there is a very clear temperature effect b. The analysis can also be done with an independent two-sample t-test, putting μ H = μ + α H and μ C = μ + α C (i) H 0 : μ H = μ C ; H 1 : not H 0 ; α = 0.05 (ii) Sample statistic: t = (Y C Y H) (μ C μ H ) ; reject for large and small values S Y C Y H BS 8 Tutorial 4

9 (iii) Under H 0 : t~t n 2 ; requirements: symmetric populations (15 n < 30) with equal variances (upper row in SPSS; ignore the result of Levene s test for a minute) (iv) t calc = ; t crit = ±2.074; p-value = Extra (note: t calc not drawn to scale) (v) reject H 0 without any doubt: there is a very clear temperature effect Why ignore Levene s test? The purpose of this exercise was to do an ANOVA with a binary variable and to compare it with an independent samples t-test. For the independent samples ttest, we have a procedure to do a test (lower line) without the equal-variance assumption, but for ANOVA we had assumed equal variances. Note that the effect is so tremendous (p = 0.000), and that even without the equal-variance assumption the null-hypothesis of no temperature effect is rejected very convincingly. Levene s test in the ANOVA case by the way also rejects equality of variance (with the same F calc and p-value): The non-parametric Kruskal-Wallis tests, by the way, give a similar pattern: and so does the independent-samples Wilcoxon-Mann-Whitney: BS 9 Tutorial 4

10 8A+B Simple regression analysis + Multiple regression analysis and other issues Q1 (based on Doane & Seward, 4/E, 12.7) a. Interpret the slope of the fitted regression HomePrice = 125, SquareFeet. b. What is the prediction for HomePrice if SquareFeet = 2,000? c. Would the intercept be meaningful if this regression applies to home sales in a certain subdivision, different form the one used to find the regression equation? Q1 a. Increasing the size of a home by 1 square foot increases the price by $150. b. HomePrice = $125,000 + ($150 2,000) = $425,000. c. The intercept might be interpreted as the value of the lot without a home. But the range of values for SquareFeet does not include zero so it would be dangerous to extrapolate for SquareFeet = 0. Extra Observe the somewhat confusing habit in economic literature of writing regression equations in the form HomePrice = 125, SquareFeet, where SquareFeet is a variable, not a unit. Q2 (based on Doane & Seward, 4/E, 12.13) The regression equation HomePrice = Income was estimated from a sample of 34 cities in the eastern United States. Both variables are in thousands of dollars. HomePrice is the median selling price of homes in the city, and Income is median family income for the city. a. Interpret the slope. b. Is the intercept meaningful? Explain. c. Make a prediction of HomePrice when Income = 50 and also when Income = 100. d. Given: R 2 = What is the meaning of that? (Data are from Money Magazine 32, no. 1 [January 2004], pp ) Q2 a. Increasing the median income by $1,000 raises the median home price by $2,610; b. If median income is zero, then the model suggests that median home price is $51,300; c. $181,800 and $312,300; d. 34% of the variance of HomePrice is explained by the model. Sol a. Increasing the median income by $1,000 raises the median home price by $2,610. b. If median income is zero, then the model suggests that median home price is $51,300. While it does not seem logical that the median family income for any city is zero, it is unclear what the lower bound would be. c. prediction HomePrice = $ (2.61 $50) = $181.8 (in $1000) or $181,800 prediction Homeprice = $ (2.61 $100) = $312.3 (in $1000) or $312,300 d. 34% of the variance of HomePrice is explained by the model. That is quite low. And it might be due to chance: perhaps a lucky sample. Fortunately, the latter can be judged by BS 10 Tutorial 4

11 statistical significance. The model is significant if the slope is significantly different from zero: this seems to be the case looking at the p-value (see later). Q3 (based on Doane & Seward, 4/E, 12.26) A regression was performed using data on 16 randomly selected charities in The variables were Y = expenses (millions of dollars) and X = revenue (millions of dollars). a. Write the fitted regression equation. b. Construct a 95 percent confidence interval for the slope. c. Perform a right-tailed t test for zero slope at α =.05. State the hypotheses clearly. (Data are from Forbes 172, no. 12, p. 248, and SUMMARY OUTPUT Regression Statistics Multiple R 0, R Square 0, Adjusted R Square 0, Standard Error 14, Observations 16 ANOVA df SS MS F Significance F Regression , , , ,07289E-08 Residual , ,13245 Total Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Intercept 7, ,0403 Revenue 0,9467 0,0936 Q3 a. Y = X; b β ; c. reject H0 Sol a. Define Y: expenses ($1,000,000); X: revenue ($1,000,000) a. Y = X b. For a 95% confidence level use t 14;0.025 = The 95% confidence interval is ± ( ) or β c. Use our 5-steps procedure: Model ( step 0 ): Y i = β 0 + β 1 X i + ε i where ε i ~N(0, σ 2 ) (i) H 0 : β 1 0 versus H 1 : β > 0 (α = 0.05) (ii) Sample statistic: B 1 ; reject for large values. (iii) Distribution test statistic under H 0 : t = B 1 β 1 ~t S n 2 B 1 Requirements: see model (iv) Calculated test statistic: t calc = = Critical value: t crit = t 14;0.05 = (note: t calc not drawn to scale) (v) Because t calc > t crit, reject H 0. There is evidence that the slope is positive; increased revenue is correlated with increased expenses. BS 11 Tutorial 4

12 Q4 Use a linear regression model to explain the height (Dutch: lengte ) of female premaster students ( ) in terms of their shoe size (Dutch: schoenmaat ). Below you find some computer output, based on a random sample of these students. Predicted values for: Lengtecm 95% Confidence Interval 95% Prediction Interval Schoenmaat Predicted lower upper lower upper a. Determine the theoretical and the estimated model belonging to the given output. b. It is claimed that the slope in this model is larger than 2. Test this hypothesis (α = 1%). c. Is this a useful model in order to predict the height of female premaster students? (Perhaps you have seen a footprint in the snow; is it useful (using this model) to predict the height of the person concerned?) d. You see a footprint of size 38 in the snow and looking up you see in the distance a (female) premaster student just walking away. Give a relevant 95% interval for the height of this (female) premaster student. BS 12 Tutorial 4

13 Sol e. The next day you see another footprint of size 38. Give a relevant 95% interval for the average height of all (female) premaster students with shoe size 38. f. Calculate a 90% confidence interval for the constant in the regression model. Q4 a. Theoretical model: Yi = β0 + β1xi + εi, with εi~n(0, σ 2 ); Estimated model: Y = b0 + b1x = X; b. reject H0; c. not very useful; d. ሾ , ሿ; e. ሾ , ሿ; f. ሾ24.467,78.305ሿ a. Theoretical model: Y i = β 0 + β 1 X i + ε i, with ε i ~N(0, σ 2 ). Y = height in cm, X = shoe size (may be stated for individual observations with or without the subscript i). Estimated model: Y = b 0 + b 1 X = X b. Use the 5 steps procedure! step 0 (model): see a. (i) H 0 : β 1 2; H 1 : β 1 > 2; α = 1% (ii) Sample statistic: B 1 ; reject for large values (iii) Distribution test statistic under H 0 : t = B 1 β 1 ~t S n 2 = t 89 (n = 91) B 1 Requirements: see model. (iv) Calculated test statistic: t calc = = Critical value: t crit = t 89;0.01 = using Excel. With the table, you may take a conservative value t crit t 85;0.01 = p-value: (using Excel) (v) Decision: do reject H 0, because p-value smaller than 1% or because t calc > t crit. Conclude that the slope is larger than 2. c. It is a statistically significant model, so the question about practically relevant is meaningful. We have R 2 = which is quite low. It would have quite limited value in predicting the height of a thief if the police found a footprint in the snow. d. This is individual prediction : Y X1 = e. This is mean prediction : Y X1 = f. b 0 ± t n 2;0.05 s B0 = ± ( ), so β (Excel: t 89;0.05 = ) Q5 A consumer products company wants to measure the effectiveness of different types of advertising media in the promotion of its products. Specifically, the company is interested in the effectiveness of radio advertising and newspaper advertising (including the cost of discount coupons). A sample of 22 cities with approximately equal populations is selected for study during a test period of one month. Each city is allocated a specific expenditure level both for radio advertising and for newspaper advertising. The sales of the product (in thousands of dollars) and also the levels of media expenditure (in thousands of dollars) during the test month are recorded, with the following results: BS 13 Tutorial 4

14 SPSS results: a. State the multiple regression equation (description of the model including assumptions and the estimated model). b. Interpret the meaning of the slopes, b 1 and b 2, in this problem. c. Interpret the meaning of the regression coefficient, b 0. d. Which type of advertising is more effective? Explain. e. Determine whether there is a significant relationship between sales and the two independent variables (radio advertising and newspaper advertising) at the 0.05 level of significance. f. Interpret the meaning of the p-value. g. Compute the coefficient of multiple determination, R 2, and interpret its meaning. h. Find the adjusted R 2 and interpret its meaning. i. Is there evidence that the slope coefficient for Radio advertisements is more than 10 at α = 0.05? BS 14 Tutorial 4

15 Sol Q5 a. Y = β0 + β1x1 + β2x2 + ε with ε~n(0, σ 2 ) and Y = b0 + b1x1 + b2x2 = X X2; d. newspaper advertising is more effective; e. there is evidence of a significant linear relationship; i. there is evidence that the coefficient for radio advertisements is larger than 10. a. Statistical model: Y = β 0 + β 1 X 1 + β 2 X 2 + ε with ε~n(0, σ^2) where Y=Sales, X 1 =Radio Advertising, X 2 =Newspaper Advertising Estimated model: Y = b 0 + b 1 X 1 + b 2 X 2 = X X 2 where Y =Estimated Sales b. For a given amount of newspaper advertising, each increase of $1000 in radio advertising is estimated to result in a mean increase in sales of $13,081. For a given amount of radio advertising, each increase of $1000 in newspaper advertising is estimated to result in the mean increase in sales of $16,795. c. When there is no money spent on radio advertising and newspaper advertising, the estimated mean amount of sales is $156, d. The slope of newspaper advertising is higher than the slope of radio advertising, so newspaper advertising is more effective. e. Model: see a. (i) H 0 : β 1 = β 2 = 0; H 1 : not H 0 ; α = 0.05 (ii) Sample statistic: F = MSR ; reject for large values MSE (iii) Under H 0 : F~F 2,19 Requirement: see model formulation (error term normally distributed with constant variance) (iv) F calc = = 40.16; F crit = 3.522; p-value = (note: F calc not drawn to scale) (v) reject H 0 because F calc > F crit or equivalently because p < α. Conclude that there is evidence of a significant linear relationship. f. p-value < : the probability of obtaining an F calc of or even larger is less than if H 0 is true. g. R 2 = SSR = = , or rather directly from SPSS output. So, 80.87% of the SST variation in sales can be explained by variation in radio advertising and variation in newspaper advertising. Note: model is significant and R 2 = 0.81, so practically it is a useful model. 2 h. R adj = from computer output. This is the proportion of explained variance, but taking into account the number of variables and number of observations. i. This test is not provided by SPSS, but is not difficult to derive from it. (i) H 0 : β 1 10 against H 1 : β 1 > 10 (α = 0.05) (ii) Sample statistic: B 1 ; reject for large values. (iii) Under H 0 : B 1 β 1 ~t S 19 ; requirements: see earlier. B 1 (iv) t calc = = ; t crit = t 19;0.05 = BS 15 Tutorial 4

16 (v) Reject H 0 because t calc > t crit. There is evidence that the coefficient for radio advertisements is larger than 10. Old exam questions Q1 (i) 22 May 2017, Q1i-1k We wish to establish a demand function Q = a bp, using measurement data in 6 different years. Data are below; some part has been erased. (j) (k) (l) Write down the estimated demand function. See (i). Give the upper bound of the 95% confidence interval for the slope coefficient for price. (2 decimals) See (i). Give the value of the usual test statistic for testing the overall significance of the model. (1 decimal) See (i). Give the value of R 2. (2 decimals) Q1 i. Q = P j k or 12.8 l Sol i. Just look at the second table, keeping in mind that the regression model uses Q = β 0 + β 1 P + ε. Of course, you may also write Q = P, with the hat symbolizing estimation. BS 16 Tutorial 4

17 j. The estimate of the slope coefficient is and its standard error Use t crit = t n 2;0.025 = 2.776, and calculate = k. Overall model significance is tested with F = MSR = = For a simple MSE regression, overall significance can also be tested with t = B 0 SE B = = l. R 2 = SSR SST = = Q2 29 March 2017, Q3 The effect of alcohol and drugs on learning achievements is a subject of intense research. A group of test subjects is asked to do a test exam, with a score between 1 (low) and 10 (high). Researchers want to find out how this relates to their use of alcohol and drugs in the week before the test was taken. For instance, student #2 reported that he used 13 alcoholic beverages and 4 times drugs in the week before the test. Results are shown below (take care: these tables use a decimal point). Some parts of the output have been suppressed. In all questions, define all symbols, except when you use standard symbols (such as R 2 ). (a) (b) (c) (d) State the the theoretical model analyzed, as well as its practical relevance (numerical) and statistical significance (numerical and the null hypothesis). Interpret practical relevance in a few words. Define all non-standard symbols. Find a 95% confidence interval for the slope coefficient for alcohol. Before taking the exam, Bob wants to relax by taking either an alcoholic beverage or a drug. Given that he likes to obtain a high grade, what can he best take: alcohol or drugs? Explain why. Test, at α = 5%, if the hypothesis that the slope coefficient for drugs is equal to 0.2 can be rejected. Use the five-step procedure. BS 17 Tutorial 4

18 Sol (a) (b) (c) exam score is explained by alcohol and drugs use. (b) CIβ1;0.95 = ሾ , ሿ. (c) Bob can best take a drug, no alcohol. (d) Do not reject H0 and conclude that there is no reason for concluding that the slope factor for drugs is unequal to 0.2. = This means that 53% of the variance in the Theoretical model: Y = β 0 + β 1 X 1 + β 2 X 2 + ε, where X 1 is the use of alcohol and X 2 is the use of drugs. Statistical significance: the p-value of this model is 0.001, with H 0 : β 1 = β 2 = 0. Practical relevance: R 2 = SSR = This means that 53% of the variance in the = SST exam score is explained by alcohol and drugs use. Use b 1 = 0.156, S B1 = and df = 18. For a 95% confidence interval we need t 18;0.025 = So, CI β1 ;0.95 = ሾ , ሿ = ሾ , ሿ. Or alternatively ± 0.101, or β Bob can best take a drug, no alcohol. One drug will decrease his expected grade by 0.147, one alcohol consumption by (d) model: see question (a) with ε~n(0, σ 2 ) i) H 0 : β 2 = 0.2; H 1 : β 2 0.2;α = 5% ii) Sample statistic: B 2 ; reject for small and large values iii) Null distribution: B 2 β 2 ~t S 18 ; requirements: see model B 2 iv) t calc = ( 0.2) = 0.726; t crit = ±t 0.025;18 = ± = SST Practical relevance: R 2 = SSR Q2 (a) Theoretical model: Y = β0 + β1x1 + β2x2 + ε, where X1 is the use of alcohol and X2 is the use of drugs. Statistical significance: the p-value of this model is 0.001, with H0: β1 = β2 = 0. v) Do not reject H 0 and conclude that there is no reason for concluding that the slope factor for drugs is unequal to 0.2. Q3 29 March 2017, Q2c A study focuses on the speed of typing WhatsThat messages, split by age group. We ask random persons in three age groups to type a standard message, and observe the time required (seconds). Sample size, mean of the typing time and standard deviation of the typing time are reported below. What is the name of the test we use to find out if the differences in mean typing times of the three age groups are significant, and what is the critical value of the usual test statistic at α = 0.05? (4 points) BS 18 Tutorial 4

19 Sol Q3 Analysis of variance, the critical value is Or: Kruskal-Wallis, the critical value is We do an analysis of variance (ANOVA). The usual test statistic is F = MSA, and under H MSW 0, we have F~F 3 1, = F 2,27. At α = 0.05, the critical value is Alternatively, you may argue for Kruskal-Wallis (non-parametric ANOVA). This uses χ 3 1 with a critical value of , BS 19 Tutorial 4