An education researcher wants to explore the relationship between the grades stu

Question

An education researcher wants to explore the relationship between the grades students receive on their First test and their Final test score in a graduate level statistics class. The following data present the First test and Final scores for ten students. The data set is listed below: [Use R and provide the R code in support of your answer. Also, provide all the details wherever necessary for full credit.]

First test score and (Final test score): 182 (285) 193 (268) 217 (310) 205 (319) 226 (334) 245 (355) 265 (372) 268 (323) 266 (407) 270 (356).

1). What is the correlation coefficient? What can you say regarding the relationship between the First test and the Final test score? Explain in details. [Use R and provide the R code]

2). Derive the least square regression line. Interpret the slope coefficient.

3). Compute the ANOVA table. Conduct the test for usefulness of this regression model. Is the First test score is useful in predicting the Final test score? Explain with all the necessary details.

4). What is estimated value of the error variance (2)?

4). What can you say regarding the goodness of fit for this regression model? Explain with any

supporting value.

5). Verify that sum of the residuals is indeed zero. [Just provide the R-code here]

6). Compute 90% confidence interval for the slope (1) coefficient. Can you perform a test of hypothesis H0 : 1 = 0 against H1 : 1 = 0 based on your computed confidence interval? Explain. Also, interpret the computed interval.

7). Provide the estimated value of the mean function E(Yh) for Xh = 187. Also, compute the estimated variance for the mean function at this particular value of Xh.

Explanation / Answer

x <- c(182,193,217,205,226,245,265,268,266,270)
y <- c(285,268,310,319,334,355,372,323,407,356)
cor(x,y)
data<- lm(y~x)
summary(data)
anova(data)

1)

cor(x,y)
[1] 0.821878

cor > 0.8

there is strong correlation between the First test and the Final test score

2)

> summary(data)

Call:
lm(formula = y ~ x)

Residuals:
    Min      1Q Median      3Q     Max
-44.608 -11.724   5.921 10.222 41.416

Coefficients:
            Estimate Std. Error t value Pr(>|t|)
(Intercept)   96.419     58.486   1.649 0.13785
x              1.012      0.248   4.081 0.00353 **
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 24.97 on 8 degrees of freedom
Multiple R-squared: 0.6755,    Adjusted R-squared: 0.6349
F-statistic: 16.65 on 1 and 8 DF, p-value: 0.003531

y^ = 96.419 + 1.012*x

3)

anova(data)
Analysis of Variance Table

Response: y
          Df Sum Sq Mean Sq F value   Pr(>F)
x          1 10378.7 10378.7 16.652 0.003531 **
Residuals 8 4986.2   623.3                  
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

4) error variance = SSE/(n-2) = MSE = 623.3

goodness of fit = R^2 = 0.6755

this means 67.55 % of variation is explained by the model

Please rate

Please post rest questions as we have to solve first 4 sub-parts only

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