Question

Be sure to clearly state your hypotheses in the hypothesis tests and state your conclusions in terms of the problem. Use ?=.?? for all tests.

The following table presents shear strength (in kN/mm) and weld diameters (in mm) for a sample of spot welds.

Diameter Strength

4.2                51

4.4                54

4.6               69

4.8                81

5.0                75

5.2               79

5.4                89

5.6               101

5.8                98

6.0               102

1. If the diameter is increased by 0.3 mm, by how much would you predict the strength to increase or decrease? (Include unit of measure.)

2. Compute a 95% confidence interval for the slope parameter, ?1

3. Can you conclude that the mean shear strength of welds with a diameter of 5.1 mm is greater than 70 kN/mm? Conduct the appropriate test of hypotheses by stating ?0 and ?1, computing the test statistic, placing bounds on the p-value, and stating your conclusion

Please show all step in the solutions

Answer 1

Here I attach the R code and Output

diameter=c(4.2,4.4,4.6,4.8,5,5.2,5.4,5.6,5.8,6)
strength=c(51,54,69,81,75,79,89,101,98,102)
fit=lm(strength~diameter)
summary(fit)

confint(fit, 'diameter', level=0.95)

The regression model is$Strength = - 67.691+28.939*diameter$1

Coefficients: Estimate std. Error t value Pr>t) (Intercept) -67.691 1 5.188 -4.457 0.00212 ** diameter 28.939 2 .959 9.7791e-05 ***

1) if the diameter is increased by 0.3 mm, then Strength = -67.691+(28.939*0.3) = - 59.0093 , the strength will increase.

2) 95% confidence interval for the slope parameter, ?1 is

> confint (fit, 'diameter', level=0.95) 2.5 % 97.5 % diameter 22.11537 35.76342

3)The mean shear strength of welds with a diameter of 5.1 mm is greater than 70 kN/mm

which is given by -67.691+(28.939*5.1)= 79.8979

Here we need to test the hypothesis of the significance of the variable diameter , that is the significance of the slope parameter, ?1

t test is used to test the hypothesis of the significance

$H_0: \beta_1=0\\ H_1: \beta_1\neq0$

From the out put the p value is 1e-05 which is < 0.05 , Hence the variable is significant

The value of the test statistic, t value is 9.779

The overall F statistic is also significant since p value is 1.002e-05< 0.05

Please comment for any doubts or clarifications else give thumps up