F23_hw2

Homework 2 (82 pts) Data is accessible from the course website: Data and Resource > Data used in class> BikeProject.csv Note: 1. In the confidence interval problems, nota that components in a confidence interval include the point estimate, the critical value, the standard error, and the margin error. The result should be computed toward the end. For example, compute a CI as 5 ± 2 ∗ 4 = 5 ± 8 =(− 3 , 13 ) 2. In computation problems, a basic rule is that you keep 3 or more significant decimal places for numbers during the working period and keep 2 or more significant decimal places at the number reported at the end. 3. In the fill-in-the-blank question, when denote or write the formula for a term, show both the general and the specific form based on the question . For example, the critical value for a one-sided t-test, H 0 μ = 0 , H a : μ > 0 is denoted by t ( 1 − α ,n − 1 ) = t ( 0.95 , 30 ) . The test statistic, t s , can be computed with a formula t s = Y s / √ n = 10 20 / √ 25 , and a value of 2.5, where the general form is t s = Y s / √ n , and the specific form is 10 20 / √ 25 , The p value can be computed with a formula Pr ( t > t s | μ 0 istrue ¿ = Pr ( t > 2.5 ,given μ 0 = 0 ) , where the general form is Pr ( t > t s | μ 0 istrue ¿ and the specific form is Pr ( t > 2.5 ,given μ 0 = 0 ) , Consider a simple linear regression Y ~ X, where X is the humidity and Y is the rental counts. The goal is to study the impact of X on Y. 1. (28) Complete the confidence interval questions. (1 pt each blank, no partial credit) a). (8) To estimate the mean response value of Y when X=0.5, the point estimate can be estimated as ^ Y h = β 0 + β 1 x h : ^ Y i = 378.88 − 303.59 ( 0.5 ) : (both the general formula and specific formula in this question)=_227.085 __(computed as this value). The standard error of this estimation is denoted _ S { ^ Y h } = s √ [ 1 n + ( x h − X ) 2 Σ ( X i − X ) 2 ] :171.66 √ [ 1 17379 + 0.016187 646.84 ] (both the general formula and specific formula in this question)=_ 1.559 _(computed as the value). At the significant level of 95%, the t-value is 1

denoted by t ( 1 − α 2 ;n − 2 ) : t ( 1 − 0.025 ; 17379 − 2 ) both the general formula and specific formula in this question)=___1.9601__(computed as this value). b). (8) To predict the single response (the next observation value), the point estimate can be estimated as __ ^ Y h = β 0 + β 1 x h : ^ Y i = 378.88 − 303.59 ( 0.5 ) : ___(both the general formula and specific formula in this question)=_ 227.085_(computed as this value). The standard error of this estimation is denoted S { Pred } ❑ = s √ [ 1 n + ( x h − X ) 2 Σ ( X i − X ) 2 + 1 ] : 171.66 √ [ 1 17379 + 0.016187 646.84 + 1 ] (both the general formula and specific formula in this question)=_171.66 _(computed as the value). At the significant level of 95%, the t-value is denoted by __ t ( 1 − α 2 ;n − 2 ) : t ( 1 − 0.025 ; 17379 − 2 ) _(both the general formula and specific formula in this question)=__1.9601___(computed as this value) c). (8) To predict the mean of m responses (the average of the next m observation values), the point estimate can be estimated as _ ^ Y h = β 0 + β 1 x h : ^ Y i = 378.88 − 303.59 ( 0.5 ) : (both the general formula and specific formula in this question)=_ 227.085 _(computed as this value, as m=3). The standard error of this estimation is denoted _ S Predmean 2 = s √ [ 1 n + ( x h − X ) 2 Σ ( X i − X ) 2 + 1 m ] :171.66 √ [ 1 17379 + 0.016187 646.84 + 1 3 ] __(both the general formula and specific formula in this question)=_ 99.12022_(computed as the value). At the significant level of 95%, the t-value is denoted by _ t ( 1 − α 2 ;n − 2 ) : t ( 1 − 0.025 ; 17379 − 2 ) _(both the general formula and specific formula in this question)=_1.644 _(computed as this value). d). (2) Answer this question without computation, when estimate the mean response value X=0.6, the corresponding standard error at is _Bigger _(bigger than/smaller than/the same as) at X=0.5 , because___you are moving farther from the center mean of the data which is farther than the expected mean allowing for more data in between to fluxuate the estimate ____. e). (2) Answer this question without computation, when estimate the mean of 10 responses, the corresponding standard error at is _Smaller __(bigger than/smaller than/the same as) estimate the mean of 3 responses at the same X level, because __As you increase sample size SE decreases ___. 2

3.(5) For a two-sided hypothesis test on the significance of the linear correlation coefficient between X and Y, H 0 : ρ = 0 ,H a : ρ≠ 0 1) (2) if a T-test is used, the test statistic is computed with the sample correlation, r , with the formula: ___ r √ n − 2 √ 1 − r 2 : 0.3229 √ 17379 − 2 √ 1 − 0.1043 _____, which is computed as _ ± 44.982____(value). 2) (1) Is this test statistic the same as the t-test in question 2a) ____YES___(Y/N). (2) Discuss when the results of the hypothesis test on the linear impact and the linear association are equivalent. They are equivlant when they are comparing it to 0. Linear impact is the effect x has on y while linear association is how well the predicted line firsts the data. There is a relationship between the two because if there is a linear impact of x and y then default there is some sort of linear association since the line is a good predictor. 4. (4) Use R to compute a 95% confidence interval for the linear correlation coefficient between Y and X. Use the confidence interval to verify the hypothesis test in 3, i.e., how to use the confidence interval to conclude the hypothesis problem in question 3; and then compare the conclusion to that in question 3). We are 95% confindent that the linear corerelation cofficeint between Bikeproject Count and Bikeproject humity is beween -0.3361 and -0.3095. This means there is a negative correlation between count and humidity. This means that the higher the humidity the lower the count of bike rentals there is. As we can see the value of r falls in the condince interveral and the t statistice is the same. 5 . (21) (Compare the ANOVA F-test and the T-test on the significance of a SLR model) We know that both ANOVA F test and T-test can be used to address the significance of the linear impact of X on Y, H 0 : β 1 = 0 , H a : β 1 ≠ 0. We have completed the T-test in the previous questions, now complete an ANOVA F-test. Note: the significance of a linear model . 4

In the simple linear regression model (SLR) with only one X, the test on the significance linear impact is equivalent to the test on the significance of the linear model . In the multiple linear regression model (MLR) with multiple Xs, the test on individual linear impact cannot imply the significance of the linear model. The significance of the linear model, H 0 : β 1 = β 2 … = β k = 0 , H a ,at least one β is different , can be tested with the Global F test, or the ANOVA F test on the entire model, F s = MSR / MSE a) (12) Do a global ANOVA F test for the significance of the SLR model. The significance of the SLR model can be defined in symbols: H 0 β 1 = 0 ,Ha : β 1 ≠ 0 Or define in the following statements. H 0 : The predictor X has no impact on Y in a linear model Y~X. H 0 : The predictor X can be dropped from the linear model Y~X. SST can be computed with the formula: _ Σ ( Y i − Y ) 2 _, and a value of _ 571761591_, the degree of freedom is computed with the formula: __n-1______, and a value of _17378_. SSE can be computed with the formula: _ Σ ( Y i − ^ Y i ) 2 _, and a value of 512143240_, the degree of freedom is computed with the formula: ___n-2___, and a value of _17377__. SSR can be computed with the formula: __ Σ ( Y − ^ Y i ) 2 __, and a value of __ 59618351__, the degree of freedom is computed with the formula: ____1_____, and a value of __17379_. b) (4) Compute the test statistic for the F test, F s . (2) The formula is ____ F s = MS R MSE = 59618351 29472.48 __ and has a value of _2022.848___. (2) How is it related to the t s in question 3a) and 3b)? This is related to t s because t s 2 = F s meaning square root of 2022.848 = 44.976 which is the t statistic. 5

2). (2) The test statistic In both tests, the test statistic is an F-statistic, which measures the ratio of the explained variance to the unexplained variance. For ANOVA F test, it measures the overall significance of the model, while for GLT F test, it focuses on the significance of specific linear combinations of predictors. 3). (4) Situations when the two methods are equivalent, and situations when only GLT T test is appropriate. Global F test is used for testing entire model and it is the same aas the GLT T test when your testing for the full model. GLT T test can be only used for Subset of a full model analysis. When your testing a hypothesis related to significance to slope coefficient both F test and GLT wil give you same reults. But GLt can only be used If you have a more complex SLR model or want to test hypotheses involving linear combinations of parameters beyond just the slope coefficient - testing whether a combination of coefficients is equal to zero- then a GLT would be more appropriate. This allows you to perform a wider range of hypothesis tests within the same modeling framework. 7