LIR 832:Mid-term Examination: Fall, 2003Answer all questions as completely as you are able. Partial credit on problems is onlypossible if I can locate arithmetic errors in your calculations. Show your work!!!Part I: Definitions: (20 points - 2 points per definition) Answer nine out of ten of the following. Provide a definition and explain the importance in statistics of each term. Do not limit yourselfto copying the formulas from the formula sheet.I. 1% Test of SignificanceA 1% test of significance is a criteria which we use to determine whether the null is to berejected. To perform this test we chose an appropriate z statistic, if our sample isgreater than 30, or an appropriate t statistic, if our sample is 30 or smaller. If the z (or tvalue) produced by our sample is greater than the critical value, we reject the null. Aninterpretation of the test is that, if our null hypothesis is true, there is no more than a 1%likelihood that we will observe a value as extreme as the 1% critical value. Alternatively,if the null were true and we ran the experiment 100 times, we would expect a value inexcess of the 1% critical value in 1 of the 100 tests.II. ExperimentAn act in which the outcome cannot be foretold with certainty. Most of life is made up ofexperiments. The potential outcomes of experiments have probabilities (or likelihoods)which indicate, ex ante, the chance that the particular outcome will occur. It is also thefrequency with which we expect a given outcome will occur in a large number ofrepeated experiments.III. Z-transformation:The z-transform permits us to convert any normally distributed random variable into astandard normal random variable and thereby use the z-table to determine theprobability of a particular outcome.IV. Sample Regression:Similar to what we learned with univariate statistics, there is a population regressionwhich indicates the relationship between the explanatory and dependent variable for theentire group we would like to know about (the population). We estimate the populationregression by taking a sample and estimating a sample regression. Similar to estimatesof population means with samples, the sample regression is affected by samplingvariability and does not exactly reproduce the population regression. Different sampleswill produce somewhat different coefficients.V. r2The coefficient of determination. is the ratio of the ESS/TSS or 1-RSS/TSS. It is ar2measure of the proportion of the variance of the dependent variable (movement of thedependent variable around its mean) which we can explain with our explanatoryvariables.VI. Representative SampleWe use samples to learn about the characteristics of the populations which we are truelyinterested in but cannot observe. For a sample to be useful, it must be representative, itmust reliably reproduce the characteristics of the population. For the most part, we usevarious forms of random sampling to produce representative samples. A problem withrandom sampling is that it is affecting by sampling variabilityVII. Independent Events:Events are independent if the occurrence of one event does not cause you to change yourexpectation about the occurence of the second event. VIII. t-statisticAlso known as student’s t, the t-distribution is used to generate critical values whensample size is 30 or less. It describes the shape of the distribution of sample means, , when sample size is under 30.xIX. Constant or Intercept in the Regression Line:The intercept is the value taken by the dependent variable when the independentvariables are all equal to zero. The intercept is fitted so that the regression lines passthrough the point of means and it assures that the average value of the residuals is zero. Although all regressions should have intercepts, they are generally not treated as beingmeaningful of themselves.X. ResidualThe difference between the actual value of the dependent variable in a regression and theprediction generated by the regression equation. The residual has its origins in theimperfect fit of the regression line. This is due to (1) omission of explanatory variablesfrom the model, (2) non-linearities in the population regression relationship, (3)measurement error and (4) randomness in behavior.Part II. Problems: Each Problem is worth 20 points. Answer five of the six problems. Ifyou chose to answer all, I will select the four with the lowest scores.A. We have five plants with identical employment located in Vermont. There is somediscussion in our firm as to whether higher earnings are associated with lesserabsenteeism (people come to work more often because they give up more if they areabsent) or higher absenteeism (they have more money so its more fun to take time off). We collect the following data on average weekly earnings and the average number ofemployees absent from each of the plants over the last year.Calculate the covariance and correlation of average weekly earnings and the numberabsent. Which view is supported by these results? You should treat this data set as asample.AverageWeeklyEarningsNumberAbsent(w-wbar)(w-wbar)^2 (a-abar) (a-abar)^2(w - w b a r ) * ( a -abar)plant A 500 120 -105 11025 40 1600 -4200plant B 750 30 145 21025 -50 2500 -7250plant C 450 100 -155 24025 20 400 -3100plant D 725 50 120 14400 -30 900 -3600plant E 600 100 -5 25 20 400 -100means 605 80sum 70500 5800 -18250(n-1) 4 4 4std dev 132.759 38.079cov(w,a)=-4562.500corr(w,a)= -0.903B. It is our belief that, as we approach winter and days become shorter, students becomedepressed if not crazed. There is a test for depression, called the slug test, whichmeasures the number of eye movements in a one minute period. Long experience withthis test shows that, in a mentally healthy population, eye movement is normallydistributed with a mean of 75 blinks per minute and a standard deviation of 10 blinks perminute. Reduction in the rate of eye movement is associated with depression,sluggishness and inexplicable homicidal rage.i. Set up a null and alternative hypothesis to test our view about the effect of shorterdays on eye movement. Explain your hypotheses.Note that these are one tailed tests.Null: eye movement in the winter is no slower than in the summerHwersummer0:intµµ≥Alternative: eye movement in winter is slower than in summerHAwersummer:intµµ<ii. Given that the population is normally distributed, should we use a z or a tstatistic?If the population is normally distributed, we can