Question

Polyester resins reinforced with fiberglass are used to fabricate wall panels of restaurants. It is theorized that adding cement kiln dust​ (CKD) to the polyester composite will increase wall panel hardness. In a​ study, hardness​ (joules [J] per squared​ centimeters) was determined for three polyester composite mixtures that used a​ 40% CKD weight ratio. The hardness values were reported as 83, 82​, and 77 J/cm^2. Research has shown that the mean hardness value of polyester composite mixtures that use a​ 20% CKD weight ratio is μ = 76 J/cm². In your​ opinion, does using a​ 40% CKD weight ratio increase the mean hardness value of polyester composite​mixtures? Support your answers statistically.

Answer 1

Null hypothesis H0: Mean hardness value of polyester composite mixtures that use a​ 40% CKD weight ratio is equal to mean hardness value of polyester composite mixtures that use a​ 20% CKD weight ratio. That is, Mean hardness value of polyester composite mixtures that use a​ 40% CKD weight ratio is equal to  76 J/cm²

Alternative hypothesis Ha: Mean hardness value of polyester composite mixtures that use a​ 40% CKD weight ratio is greater than mean hardness value of polyester composite mixtures that use a​ 20% CKD weight ratio. That is, Mean hardness value of polyester composite mixtures that use a​ 40% CKD weight ratio is greater than 76 J/cm²

Reported hardness values are 83, 82​, and 77 J/cm^2.

Sample mean hardness = (83 + 82 + 77) / 3 = 80.67

Sample standard deviation, s = = 3.21455

Standard error of sample mean = s / = 3.21455 / = 1.856

Test statistic = (Observed mean - Hypothesized mean) / standard error

= (80.67 - 76) / 1.856

= 2.52

Degree of freedom = n-1 = 3 - 1 = 2

P-value = P(t > 2.52, df = 2) = 0.064

If we use the significance level = 0.05,

Since p-value is greater than 0.05 significance level, we fail to reject null hypothesis H0 and conclude that there is no significant evidence that mean hardness value of polyester composite mixtures that use a​ 40% CKD weight ratio is greater than mean hardness value of polyester composite mixtures that use a​ 20% CKD weight ratio. Thus, there is no significant evidence 40% CKD weight ratio increase the mean hardness value of polyester composite​ mixtures.

If we use the significance level = 0.10,

Since p-value is less than 0.05 significance level, we reject null hypothesis H0 and conclude that there is significant evidence that mean hardness value of polyester composite mixtures that use a​ 40% CKD weight ratio is greater than mean hardness value of polyester composite mixtures that use a​ 20% CKD weight ratio. Thus, there is significant evidence 40% CKD weight ratio increase the mean hardness value of polyester composite​ mixtures.