Problem 1. Bo + Bxi +ûi. Prove: Given data (yi, i) for i = 1,...,n, we run a simple linear regression Y₁ = Σûixi = 0. 2
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- Olympic Pole Vault The graph in Figure 7 indicates that in recent years the winning Olympic men’s pole vault height has fallen below the value predicted by the regression line in Example 2. This might have occurred because when the pole vault was a new event there was much room for improvement in vaulters’ performances, whereas now even the best training can produce only incremental advances. Let’s see whether concentrating on more recent results gives a better predictor of future records. (a) Use the data in Table 2 (page 176) to complete the table of winning pole vault heights shown in the margin. (Note that we are using x=0 to correspond to the year 1972, where this restricted data set begins.) (b) Find the regression line for the data in part ‚(a). (c) Plot the data and the regression line on the same axes. Does the regression line seem to provide a good model for the data? (d) What does the regression line predict as the winning pole vault height for the 2012 Olympics? Compare this predicted value to the actual 2012 winning height of 5.97 m, as described on page 177. Has this new regression line provided a better prediction than the line in Example 2?If your graphing calculator is capable of computing a least-squares sinusoidal regression model, use it to find a second model for the data. Graph this new equation along with your first model. How do they compare?Find the equation of the regression line for the following data set. x 1 2 3 y 0 3 4
- Repeat Example 5 when microphone A receives the sound 4 seconds before microphone B.Respiratory Rate Researchers have found that the 95 th percentile the value at which 95% of the data are at or below for respiratory rates in breath per minute during the first 3 years of infancy are given by y=101.82411-0.0125995x+0.00013401x2 for awake infants and y=101.72858-0.0139928x+0.00017646x2 for sleeping infants, where x is the age in months. Source: Pediatrics. a. What is the domain for each function? b. For each respiratory rate, is the rate decreasing or increasing over the first 3 years of life? Hint: Is the graph of the quadratic in the exponent opening upward or downward? Where is the vertex? c. Verify your answer to part b using a graphing calculator. d. For a 1- year-old infant in the 95 th percentile, how much higher is the walking respiratory rate then the sleeping respiratory rate? e. f.3. The research shows that both X₁ and X₂ are theoretically valid, meaning that they both belong in the true regression relationship. Yi Bo + B₁X1i + B₂X₂i + Ei The researcher decides to estimate two regressions for the sample: Y₁ = Bo + B₁X₁i + Ei Y₁ = Bo + B₂X2i + €₁ Would the estimated regression coefficients for ₁ and ₂ be the same as if the researchers had estimated the true regression relationship? How will the error term compare between models? How will the R-squared values for these two regressions compare to the R-squared of the true relationship? Explain and support your answers with relevant formulas.
- A) A linear regression has a =6 and b=5 what is y predicted as when x=9? B) A linear regression has b=3 and a=4.What is the predicted Y for x=7?Suppose that you have the following three regression models: Yi B₁ + B₂x₁ + Uj regression) Y₁ = a₁ + a₂x₁ + eį, if i is male male individuals) Yi = ₁ + ₂x₁ + Ei, if i is female (separate regression for female individuals) ESS RSS The error terms satisfies that E[ui] = E[ei] = E[ei] = 0. Then, we find the following RSS and ESS: Pooled Regression 124.3 25 (separate regression for (pooled 71.1 11.4 separate separate regression for regression for male individuals female individuals 56.6 8.6 The number of sample size is given by 18. Select the correct statement(s). The RSS from the pooled regression is always smaller than the sum of RSSS from the separate regression models. In the above setup, Chow test is equal to 2. If e; is correlated with €;, then the OLS estimator of a2 from the separate regression will be is biased. If the variances of e; is different from €;, then the OLS estimator of 3₂ is no longer unbiased estimator. If we add another variable z; to the pooled regression, then the…In Step 2: Construct an estimated simple linear regression model how did you come up with the column X*X ?
- 4. In multiple regression, why do we prefer our IVs to be uncorrelated with each other (i.e., we want rx, x, to be 0)?5. For two variables x and y with the same mean, the two - a regression equations are y= ax + b and x= ay + B. Show that 1- a Find also the common mean.In (WAGE) = B, + B,EDUC + B,EDUC² + B,EXPER+ B,EXPER² + Bo (EDUC × EXPER) + e The above regression is fitted with a range of restrictions. The regression results are given below: Coefficient Estimates and (Standard Errors) Eqn (A) Eqn (C) Eqn (D) Eqn (E) Variable Eqn (B) 0.403 1.483 1.812 2.674 1.256 (0.771) (0.495) (0.494) (0.109) (0.191) EDUC 0.175 0.0657 0.0669 0.0997 (0.091) (0.0692) (0.0696) (0.0117) EDUC -0.0012 0.0012 0.0010 (0.0027) (0.0024) (0.0024) EXPER 0.0496 0.0228 0.0314 0.0222 (0.0172) (0.0091) -0.00038 -0.00032 (0.0104) (0.0090) EXPER -0.00060 -0.00031 (0.00019) (0.00019) (0.00022) (0.00019) EXPER X EDUC -0.001703 (0.000935) SSE 37.326 37.964 40.700 52.171 38.012 What is the value of F-test statistic for the restriction on the coefficients of Equation E that give Equation D? Note: Sample size N is 200 Select one: O a. 3.89 O b. 5.85 O. 1.78 O d. 73.01
