Question

1. An article in Wood Science and Technology reported a study of the deflection (mm) of particleboard from stress levels of relative humidity. Assume that the two variables are related to the simple linear regression model. Let x=stress level % and y=deflection mm

x	54	54	61	61	68	68	75	75	75
y	16.47	18.69	14.3	15.12	13.51	11.64	11.17	12.53	11.22

1. What is the estimated linear regression equation?                                                                                   

2. Let β be the coefficient of x in the linear regression equation between x and y. Write down the null and alternative hypotheses to test if the regression is significantly negative.                                                 

3. Test the hypotheses written in (c) using the critical value approach at α=5%. You must clearly give the test statistic, critical value, rejection region, decision on H0 and the interpretation of the test result.

  

4. Write down the null and alternative hypotheses to test if the slope β is significantly greater than -0.1. Test the hypotheses using an appropriate confidence interval. Interpret the test result.          

5. Write down the null and alternative hypotheses to test if the slope β is significantly lesser than -0.1. Test the hypotheses using an appropriate confidence interval. Interpret the test result.   

Answer 1

In order to solve this question I used R software.

R codes and output:

> x=c(54,54,61,61,68,68,75,75,75)

> y=c(16.47,18.69,14.3,15.12,13.51,11.64,11.17,12.53,11.22)

> fit=lm(y~x)

> summary(fit)

Call:

lm(formula = y ~ x)

Residuals:

     Min       1Q   Median       3Q      Max

-1.56361 -0.61194 -0.04444 0.30639 1.60806

Coefficients:

            Estimate Std. Error t value Pr(>|t|)   

(Intercept) 32.04123    2.88431 11.109 1.07e-05 ***

x           -0.27702    0.04359 -6.355 0.000384 ***

---

Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 1.057 on 7 degrees of freedom

Multiple R-squared: 0.8523, Adjusted R-squared: 0.8312

F-statistic: 40.38 on 1 and 7 DF, p-value: 0.0003836

a.

Estimated linear regression equation:

Y = 32.04123 - 0.27707 X

b.

Hypothesis:

HO: BO H:8< 0

c.

Test statistic,

$\\ t = \frac{\hat{\beta_1 } - \beta_1 }{S.E(\hat{\beta_1})} \\ \\ \\ t = \frac{-0.27707 - 0}{0.04359} \\\\ t = -6.3563$

Degrees of freedom = n-1 = 8-1 = 7

Critical value = -1.895

Rejection region = Region below -1.895

Since calculated value fall in the rejection region, we reject null hypothesis . And conclude that slope coefficient is significantly negative.

d.

Hypothesis:

$\\ H_0 : \beta \leq -0.1 \\ H_a: \beta > -0.1$

95 % confidence interval:

$\\ C.I. = [\hat{\beta }_1 - t_{\alpha /2,n-2} *SE(\hat{\beta }_1) , \infty ]\\\\ C.I. = [-0.27707 - 1.895 * 0.04359, \infty] \\ \\ C.I. = (-0.3597, \infty)$

Since above confidence contain values less than -0.1 also, hence we accept null hypothesis and conclude that slope β is significantly lesser than -0.1.

e.

Hypothesis:

$\\ H_0 : \beta \geq -0.1 \\ H_a: \beta < -0.1$

95 % confidence interval:

$\\ C.I. = [ -\infty \ , \ \hat{\beta }_1 + t_{\alpha /2,n-2} *SE(\hat{\beta }_1) ]\\\\ C.I. = [ -\infty, \ -0.27707 + 1.895 * 0.04359] \\ \\ C.I. = (- \infty, \ -0.1945)$

Since above confidence contain values less -0.1, hence we reject null hypothesis and conclude that slope β is significantly lesser than -0.1.