Closed book, notes and no electronic devices. 10 points per correct answer, 20 points for signing your name.

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1 Quiz 1. Name: 10 points per correct answer, 20 points for signing your name. 1. Pick the correct regression model. A. Y = b 0 + b 1X B. Y = b 0 + b 1X + e C. Y X = x ~ p(y x) D. X Y = y ~ p(y x) 2. How is prediction different from forecasting? A. Prediction requires the normality assumption, forecasting does not B. Prediction can refer to the past, forecasting always refers to the future C. Prediction uses probabilistic models, forecasting uses deterministic models D. There is no difference between the two 3. The distribution p(y x) refers to A. A population of Y values B. A sample of Y values C. Potentially observable Y data D. Actually observed Y data 4. The experiment where a student was allowed to pick where they stand before attempting to throw the ball in the trash can was a experiment. A. Fixed-X B. Random-X C. randomized D. pre-test / post-test 5. The data set containing students GPAs and GMAT scores was separated into two groups. What were those groups? A. Males and Females B. Masters and Ph.D. students C. Business and non-business majors D. High GPA and low GPA students 6. What is a statistical model? A. A recipe for producing random data B. Something you calculate from data C. A deterministic equation 7. When is the linearity assumption in regression valid? A. When there are three or more levels of the X variable B. When the test for linearity passes (p >.05) C. When Y increases as X increases D. None of the above 8. What does LOWESS estimate? A. the conditional mean function B. the conditional variance function C. the conditional distribution function D. the autocorrelation function

2 Quiz 2. Name: 9. Data (xi, yi) are produced by the classic model Y = β0 + β1 X + ε, where ε ~ N(0, σ 2 ). Which line minimizes the sum of squared errors (SSE)? A. The line β0 + β1 x, where β0 and β1 are the true values of the parameters B. The line ˆ β ˆ 0 + β 1 x, where 0 ˆβ and ˆβ 1are the maximum likelihood estimates assuming the classic model C. The line 0 + 1x D. The line 1 + 0x 10. When should you use maximum likelihood estimates based on the Laplace distribution? A. When your distributions p(y x) are skewed B. When your distributions p(y x) are outlier-prone C. When your distributions p(y x) are discrete D. When your distributions p(y x) are normal 11. What distribution p(y x) is assumed in the Gauss-Markov Theorem? A. Gaussian B. Markovian C. Laplacian D. No particular distribution is assumed 12. What does the Gauss-Markov Theorem tell you? A. OLS is better than ML B. ML is better than OLS C. OLS is best among linear, unbiased estimators D. ML is best among linear, unbiased estimators

3 Quiz 3. Name: 1. In the discussion of 1 as an unbiased estimator of β 1, what is true? A. ˆβ 1 is random, and β 1 is random B. ˆβ 1 is fixed, and β 1 is random C. ˆβ 1 is random, and β 1 is fixed D. ˆβ 1 is fixed, and β 1 is fixed 2. If ˆβ 1 is an unbiased estimator of β 1, then A. ˆβ 1 is equal to β 1 B. ˆβ 1 is sometimes equal to β 1 C. ˆβ 1 is close to β 1 D. None of the above 3. Recall that the errors are ε i = {Y i (β 0 + β 1 x i)}, and the residuals are e i = {Y i ( ˆβ ˆβ x i)}. Pick the true statement about these terms, as they apply to the real world (not simulation). A. The {ε i} are observable, and the {e i} are observable B. The {ε i} are not observable, and the {e i} are observable C. The {ε i} are observable, and the {e i} are not observable D. The {ε i} are not observable, and the {e i} are not observable 4. Your estimated ˆβ 1 is equal to 3.4, with a standard error of 1.2. What is 1.2? A. The standard deviation of 3.4. B. The estimated standard deviation of 3.4 C. The standard deviation of potentially observable values of ˆβ 1 D. The estimated standard deviation of potentially observable values of ˆβ 1

4 Quiz 4. Name: 1. The exact 95% confidence interval for β1 in the Toluca case (lotsize, workhours) used the critical from the T distribution with 23 degrees of freedom. That critical value was A B. 2.0 C Consider the Toluca case, where X = lotsize and Y = workhours. The expression refers to E(Y X = 20) A. the actual workhours that will be observed on a single job where lotsize = 20. B. the estimate of the workhours for a single job where lotsize = 20 C. the estimate of the mean of all potentially observable workhours for jobs where lotsize = 20 D. the true mean of all potentially observable workhours where lotsize = Under the classic model, what happens to the prediction interval for Y X = x when n increases? A. It gets wider B. It approaches β0 + β1x ± 0 C. It approaches β0 + β1x ± 1.96σ 4. You get a p-value that is p = 0.57 for testing whether a particular β = 0. Then A. the difference between ββ and 0 is explainable by chance alone B. the difference between β and 0 is explainable by chance alone C. the true β is close to zero D. the true β is equal to zero

5 Quiz 5. Name: 5. According to the author, what should be your first step in the analysis of any regression data? A. Plot the (xi, yi) data B. Determine whether the difference between ββ and 0 is explainable by chance alone C. Find the R 2 statistic D. Calculate summary statistics (means, standard deviations, etc.) for the X data and the Y data 6. When checking assumptions using a hypothesis testing method, the null hypothesis always states A. The assumption is precisely true B. The assumption is close enough to being true C. The assumption is false D. The assumption may be false, or it may be true 7. Which method is the preferred method to check assumptions? B. Use a test for the assumption, and conclude that the assumption is true if p >.05 C. Use a test for the assumption, and conclude that the assumption is true if p <.05 D. Use an appropriate graph to check the assumption E. Use a literature search to see what other researchers say about the assumption 8. What does the linearity assumption state? A. The data Y fall on a straight line, β0 + β1x B. The predicted values fall on a straight line, β0 + β1x C. The means of the potentially observable Y values fall on a straight line, β0 + β1x D. The means of the actually observed Y values fall on a straight line, β0 + β1x

6 Quiz 6. Name: 1. If there is homoscedasticity in the Toluca case (X = lotsize, Y = workhours), what is true? A. Var(Y X = 20) = Var(Y X = 120) B. Var(Y X = 20) is close to Var(Y X = 120) C. Var(Y X = 20) is far from Var(Y X = 120) D. Var(Y X = 20) may be far from or close to Var(Y X = 120); you can t tell from the given information which is true. 2. To test for heteroscedasticity, you test the null hypothesis that the slope of a certain regression is zero. Which regression do you use for this test? A. The regression of ei on xi. B. The regression of yi on xi C. The regression of ei on yi D. The regression of ei on yy ii 3. To test for autocorrelation, you test the null hypothesis that the correlation between two variables is zero. Which variables? A. Yt and Yt-1 B. Xt and Xt-1 C. εt and εt-1 D. βt and βt-1 4. To test for non-normality, you apply the Shapiro-Wilk test (shapiro.test) to which data? A. The response variable, Y B. The predictor variable, X C. The predicted values, yy D. The residuals, e

7 Quiz 7. Name: 1. What problems can be lessened by transforming X only (and not Y)? Select all that apply. Ten points per correct selection/non-selection. A. Non-randomness B. Non-normality of the distributions p(y x) C. Non-constant variance (heteroscedasticity) D. Non-linearity (curvature) E. Poor predictive ability (low R 2 ) F. Outliers in the X data 2. Suppose U = ln(y). Here is an estimated regression function: U ˆ = X. Back-transform this function to give the relationship between Y ˆ and X.

8 Quiz 8. Name: 3. What was the problem with comparing likelihoods to select a better transformation of Y? A. Since the distribution of the back-transformed Y is non-normal, there is no likelihood. B. You can t compare the likelihood for Y with the likelihood for a transformed Y. You have to compare likelihoods using the same Y variable. C. Since one of the models will be non-linear, likelihood is meaningless. D. The likelihoods are grossly affected by outliers, so you should not use the likelihood for an untransformed model when there are outliers. 4. Suppose Y has measurement units thousands of dollars per job. What are the measurement units of 1/Y? 5. Given X = x, the Box-Cox method assumes what about the transformation Y λ? A. Y λ is normally distributed B. Y λ is log-normally distributed C. Y 1/λ is normally distributed D. Y 1/λ is log-normally distributed 6. Elasticity, as estimated by the regression of ln(y) on ln(x), refers to A. change in Y per unit change in X B. change in the median of Y per unit change in X C. percentage change in Y per 1% change in X D. percentage change in the median of Y per 1% change in X

9 Quiz 9. Name: 7. Suppose E(Y X = x) = 3 + 4x. Suppose E(X) = 10. Then E(Y) = 8. See problem 1. Given X = x, the best predictor of Y is 9. The Law of Total Variance tells us that Var(Y ) = Var{E(Y X)} + E{Var(Y X)}. Suppose Var{E(Y X)} = 30 and E{Var(Y X)} = 10. Then the true R 2 is 10. The relationship between X = ice cream sales and Y = drownings was shown to be A. explained by chance alone B. explainable by chance alone C. predictive D. causal

10 Quiz 10. Name: 1. (20) Consider the model GPA {GMAT = x1, PHD = x2} ~ N(β 0 + β 1x1 + β 2x2, σ 2 ) of the reading. If this model is true, then the coefficient β 2 is equal to A. The difference between GPAs for Ph.D. and Master s students. B. The difference between mean GPAs for Ph.D. and Master s students. C. The difference between GPAs for Ph.D. and Master s students who have the same GMAT. D. The difference between mean GPAs for Ph.D. and Master s students who have the same GMAT. E. The difference between GPAs for Ph.D. and Master s students, all else held fixed. F. The difference between mean GPAs for Ph.D. and Master s students, all else held fixed. 2. (20) Assuming the same model as in 1 above, σ is the A. standard deviation of the GPAs. B. standard deviation of GMATs. C. standard deviation of PHDs. D. standard deviation of the GPAs for potentially observable PHD students who have GMAT score = (40) Let A be a 3x3 matrix. Then A -1 A = (give the explicit matrix.)

11 Quiz 11. Name: 1. With two X variables, X1 and X2, the least squares regression fit (in R, lm(y ~ X1+X2)) gives you A. the line that minimizes the sum of squared deviations from the X s to the line. B. the line that minimizes the sum of squared deviations from the Y s to the line. C. the plane that minimizes the sum of squared deviations from the X s to the plane. D. the plane that minimizes the sum of squared deviations from the Y s to the plane. 2. Which one is the matrix form of the regression model? A. Y = Xβ B. Y = Xβ + ε C. ˆβ = (X T X)(X T Y) D. ˆβ = (X T X) 1 (X T Y) 3. The classical regression model assumes what about the conditional covariance matrix of the potentially observable Y data? Cov(Y X = x) = A. I B. (x x) -1 C. σ 2 I D. σ 2 (x x) In the simulation of the reading there were two models to predict a common Y, one using an actual X (called XA), and the other using a guess of X (called XG). The true models have coefficients βa and βg, respectively; their estimates were ˆA β and ˆG β, respectively. The conclusions were: A. ˆG β was an unbiased estimate of βg, and ˆG β was an unbiased estimate of βa B. ˆG β was an unbiased estimate of βg, but ˆG β was a biased estimate of βa C. ˆG β was a biased estimate of βg, but ˆG β was an unbiased estimate of βa D. ˆG β was a biased estimate of βg, and ˆG β was a biased estimate of βa

12 Quiz 12. Name: 1. What is the point of using the adjusted R square statistic? A. It is equal to the population R 2 statistic B. It is less biased than the usual R 2 statistic C. It measures causal effects of the X variables D. It is less sensitive to violations of the assumptions 2. A multiple regression model is Y = β0 + β1x1 + β2x2 + ε. The null hypothesis that is tested by the F test is A. H0 : β0 = β1 = β2 = 0 B. H0 : β1 = β2 = 0 C. H0 : β0 = β1 = β2 D. H0 : β1 = β2 3. What happens to the distribution of the F statistic when the null hypothesis is false? A. It is shifted to the left of the usual F distribution B. It is shifted to the right of the usual F distribution C. It becomes a bimodal distribution D. It becomes a degenerate distribution 4. Your regression model is Y = β0 + β1x1 + β2x2 + ε. Multicollinearity in your regression model is a concern when R 2 is close to 1.0 for which of the following? (Select only one.) A. lm(y ~ X1) B. lm(y ~ X2) C. lm(y ~ X1 + X2) D. lm(x1 ~ X2)

13 Quiz 13. Name: 1. A fitted quadratic model is yy = 4 3x + 2x 2. The value of x that gives the minimum yy is A B. 2.0 C. 3.0 D When graphed in three dimensions, the function f (x1, x2) = b0 + b1x1 + b2x2 + b3x1x2 looks like A. A plane B. A line C. A twisted plane D. A bell curve 3. In the model of 2 above, the coefficient b1 is A. the effect of X1 when X2 is held fixed. B. the effect of X1 when all else is held fixed. C. the effect of X1 regardless of X2. D. the effect of X1 when X2 is equal to In the model of 2 above, the variable inclusion principle states that you should A. include X1 in the model whenever you include X1X2 in the model. B. include X1 in the model only when X1 is statistically significant. C. include X1 in the model only when X1X2 is statistically significant. D. include X1 in the model only when the intercept is statistically significant.

14 5. The model to predict Grade Point Average (GPA) from the PhD indicator variable (1=Ph.D., 0=Masters) was GPA = β0 + β1phd + ε. In this model the linearity assumption was A. Certainly true B. Certainly false C. True only if β1 was insignificant D. True only if β1 was significant 6. See 5 above. What happens when you code the GPA variable as (0=Ph.D., 1=Masters) instead of (1=Ph.D., 0=Masters)? A. Nothing at all B. The β coefficients change C. The R 2 changes D. The constant variance assumption needs to be re-evaluated 7. In the housing price example, there were five regions (A,B,C,D, and E) and the indicator variable model was Price = β0 + β1xa + β2xb + β3xc + β4xd + ε. The parameter β1 is A. The price in region A B. The mean price in region A C. The price in region A, minus the price in region E D. The mean price in region A, minus the mean price in region E 8. In the housing price example of quiz question 7 above, you can bring square footage of the home into the model in a way that allows different effects of square footage for different regions. How do you do this? A. Incorporate interaction terms between square footage and region indicators B. Perform simulation study to see whether square footage and region interact C. Test for simultaneous significance of square foot and region effects D. Draw a graph of the distribution of price when square footage and region are held fixed

15 Quiz 14. Name: Consider a two-way ANOVA to predict price of a home as a function of region (A or B) and whether the home has a cellar (Yes or No). The model is Price = β0 + β1region.a + β2cellar.yes + ε Region.A = 1 if the home is in Region A, and Region.A = 0 if the home is in region B. Cellar.Yes = 1 if the home has a cellar, and Cellar.Yes = 0 if the home does not have a cellar. Fill in the table of mean values, as shown in the book. Table 1. True mean values of Price, according to the model Price = β0 + β1region.a + β2cellar.yes + ε. Cellar No Cellar Region A Region B

16 Quiz 15. Name: In the book, two models were analyzed: fit1 = lm(charity ~ INCOME + DEPS) fit2 = lm(charity ~ INCOME + as.factor(deps)) Recall: CHARITY is a measure of charitable contributions. INCOME is a measure of income. DEPS is number of dependents claimed on a tax form, taking the values 0, 1, 2, 3, 4, 5 and 6 in the data set. How are the theoretical models represented by fit1 and fit2 different?

17 Quiz 16. Name: 1. Hans had X1 = GRE quant = 140, X2 = GRE verbal = 160, and X3 = Undergrad GPA = 2.7. Which number is likely to be closest to Hans finishing GPA? A. E(GPA) B. E(GPA X1 = 140) C. E(GPA X1 = 140, X2 = 160) D. E(GPA X1 = 140, X2 = 160, X3 = 2.7) 2. See question 1. What is E(GPA X1 = 140)? A. The average of the GPAs among students in the sample who have GRE quant = 140 B. The average of potentially observable GPAs when GRE quant = 140 C. The predicted GPA using the estimated regression model with X1 = 140 D. Hans GPA, assuming all we know about him is his GRE quant score 3. Three different regression models can be estimated using data to predict Hans GPA: fit1 = lm(gpa ~ GRE.quant) fit2 = lm(gpa ~ GRE.quant + GRE.verbal) fit3 = lm(gpa ~ GRE.quant + GRE.verbal + GPA.undergrad) Each of these models gives prediction of Hans GPA by plugging in the appropriate X values. Which model gives a prediction having highest variance? A. fit1 B. fit2 C. fit3 D. because of homoscedasticity, all variances are the same 4. What does the variance-bias tradeoff tell you? A. Unbiased estimates are always best B. Estimates with smaller variance are always best C. Biased estimates can be better when their variance is small D. Biased estimates can be better when their variance is large

18 Quiz 17. Name: 1. The BIC statistic is BIC = 2(Log Likelihood) + {ln(n)} (# of estimated parameters). When using this statistic for variable selection, you should choose a set of variables for which BIC is A. Close to 1.0 B. Close to 0.0 C. Large D. Small 2. Suppose the variance function is Var(Y X = x) = γ 2 x 2. Using this function, the maximum likelihood estimates are weighted least squares estimates, with weights equal to A. x B. x 2 C. 1/x D. 1/x 2 3. When is WLS more efficient than OLS? A. When all the assumptions are satisfied for the classical regression model B. When the distributions p(y x) are non-normal distributions C. When the variance function is correctly specified D. When the variance function is not correctly specified 4. Why is it better to assume (i) σ(y X = x) = exp(γ0 + γ1x), rather than to assume (ii) σ(y X = x) = γ0 + γ1x? A. Because standard deviations cannot be negative B. Because standard deviations are normally distributed C. Because the log likelihood is higher for (i) D. Because (i) is true and (ii) is false

19 Quiz 18. Name: 1. What assumption do you make about σ(y X = x) when you use the heteroscedasticityconsistent standard errors? A. σ(y X = x) = γx B. σ(y X = x) = γ0 + γ1x C. σ(y X = x) = exp(γ0 + γ1x) D. No assumption regarding the relationship between σ(y X = x) and x is made at all 2. When you use the heteroscedasticity-consistent standard errors, what is the estimate of 2 σ 2 = Var(Y2 X2 = x2)? Α. {y2 (β0 + β1x2)} 2 2 B. { y ( ˆ β + ˆ β x )} C. RMSPE D. No estimate of σ 2 = Var(Y2 X2 = x2) is used at all 3. (40) What is different about Cov(Y X = x) when the observations are correlated versus when they are uncorrelated? (Be brief).

20 Quiz 19. Name: 1. What distribution do you assume for p(y x) when you use logistic regression? A. Normal B. Logistic C. Multinomial D. Bernoulli 2. What distribution do you assume for p(y x) when you use multinomial logistic regression? B. Normal B. Logistic C. Multinomial D. Bernoulli 3. To estimate parameters of the logistic regression model you use, and to estimate the parameters of the multinomial logistic regression model you use. A. Least squares, Least squares B. Least squares, Maximum likelihood C. Maximum likelihood, Least squares D. Maximum likelihood, Maximum likelihood 4. If the probability is 0.25, then the odds ratio is A. 1/4 B. 4.0 C. 1/3 D. 3.0

21 Quiz 20. Name: 5. What is a latent variable? A. A variable in your data set B. A skewed variable C. An unobserved variable D. A multicollinear variable 6. What distribution was assumes for the latent variable (called Z ) in the reading? A. Normal B. Logistic C. Multinomial D. Bernoulli 7. The multinomial logistic regression model A. has the same number of parameters as the ordinal regression model B. has fewer parameters than the ordinal regression model C. has more parameters than the ordinal regression model 8. In the Boss rating example, where Y could be 1 (Low), 2 (Medium), or 3 (High), and X = salary, it was shown that Pr(Y = 1 X = x) A. increases as X increases B. decreases as X increases C. does not change as X increases

22 Quiz 21. Name: 9. Give the complete list of possible values of Y, according to the Poisson regression model. A. 0, 1, 2, 3, B. 1, 2, 3, C. 0,1 D. all numbers between - and 2. Suppose you have a data set with a single Y and a single X variable. Suppose also that there are 10 distinct values of your Y variable in your data set. You are considering different regression models. Which one will have the most parameters? A. Classical B. Ordinal C. Poisson D. Negative binomial 3. Which is evidence of extra-poisson variation? A. The mean is greater than the median B. The mean is less than the median C. The mean is greater than the variance D. The mean is less than the variance 4. How do data from the negative binomial (NB) distribution differ from data from the Poisson distribution, assuming both distributions have the same mean? A. There is a larger probability of seeing large values (outliers) when data come from NB B. There is a smaller probability of seeing large values (outliers) when data come from NB C. There is a larger probability of seeing negative numbers when data come from NB D. There is a smaller probability of seeing negative numbers when data come from NB

23 Quiz 22. Name: Give an example of a censored observation. Your words should be clear as to why that observation is called censored. Be brief.

24 Quiz 23. Name: 1. When you raise 0.7 to the 0.9 power, you get A B C. 1.6 D Suppose, in the time until divorce example, that S(20 Educ = > 15 years ) = Then A. 75% of marriages have husband education greater than 15 years B. 25% of marriages have husband education greater than 15 years C. 75% of couples whose husband has greater than 15 years education are married at least 20 years D. 25% of couples whose husband has greater than 15 years education are married at least 20 years 3. In the Tobit analysis of back injury data, what was the recommendation for dealing with the outlier? A. Remove it B. Since it was a mistake, to replace it with the correct value C. Replace it with a smaller value D. Use a different distribution p(y x) 4. What was the ugly rule of thumb for identifying an outlier, as presented in the book? A. An observation that is less than two standard deviations from the mean is an outlier B. An observation that is more than two standard deviations from the mean is an outlier C. An observation that is less than three standard deviations from the mean is an outlier D. An observation that is more than three standard deviations from the mean is an outlier

25 Quiz 24: Name Closed book, notes and no electronic devices Choose one answer only for each. 1. Which statistic measures outlier in X space? A. Leverage B. Standardized residual C. Cook s distance D. z = ( y y)/ s y i y 2. Which statistic measures outlier in Y X space? B. Leverage B. Standardized residual C. Cook s distance D. z = ( y y)/ s y i y 3. Which statistic measures outlier in both Y X space and in X space? C. Leverage B. Standardized residual C. Cook s distance D. z = ( y y)/ s y i y 4. Which statistic measures outlier in Y space? D. Leverage B. Standardized residual C. Cook s distance D. z = ( y y)/ s y i y

26 Quiz 25: Name Closed book, notes and no electronic devices n 1. Let SAE(b0) = yi b0. When is there a unique value b0 that minimizes SAE(b0)? i= 1 A. When the second derivative of SAE(b0) is negative B. When the second derivative of SAE(b0) is positive C. When n is an even number D. When n is an odd number 2. Consider the classical regression model Y X = x ~ N(β0 + β1x, σ 2 ). Assuming this model is true, the function that relates the median of the distribution of Y to X = x is y0.5 = A. β0 + β1x 0.5σ B. β0 + β1x C x D x 3. What is a Winsorized data value? A. A data value that has been deleted B. A data value that has been standardized C. A data value that has been replaced with a less extreme data value D. A data value that has been replaced with a more extreme data value 4. The simulation studies in the reading about Winsorization showed that A. Winsorization gives the most powerful tests B. Winsorization gives the most accurate estimates C. Winsorization gives the most accurate prediction intervals D. you should not use Winsorization

27 Quiz 26: Name Closed book, notes and no electronic devices 5. (20) What does tree refer to in tree regression? A. The assumption of a tree distribution p(y x) B. The graphical output of the analysis looks like an upside-down tree C. The assumed conditional mean function is continuous, like branches of a tree D. The person who invented was named Tree. 6. (20) When a node of a tree produced by rpart is given as X >= 1.5 then the left branch under the node corresponds to A. X >=1.5 B. X < The neural network regression models discussed in the book are (select all that apply; eight points per correct selection/nonselection) A. universal approximators B. continuous functions C. polynomial functions D. trigonometric functions E. nonlinear functions

28 Quiz 27: Name Closed book, notes and no electronic devices 8. According to the authors, what is a regression model? A. A model to predict a Y variable, given values of X variable(s) B. A model for the conditional mean of a Y variable, given values of X variable(s) C. A model for the conditional distribution of a Y variable, given values of X variable(s) D. A model for the conditional variance of a Y variable, given values of X variable(s) 9. What is p hacking? A. An acceptable scientific practice B. Trying different methods until you get a p-value that supports your theory C. Use of simulated data sets to clarify your theory D. Finding the best probability model using different logistic regression functions 10. Which methods are most similar to likelihood-based methods? A. Heteroscedasticity consistent B. Bayesian C. Quantile based D. Method of Moment based 11. What is the recommended Step 1 of regression analysis, according to the authors? A. Specify your conditional mean function B. Specify your conditional distributions C. Specify your conditional variance function D. Look at your data