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