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Alex787 [66]
2 years ago
15

The strength of paper used in the manufacturing of cardboard boxes (y) is related to percentage of hardwood concentration in the

original pulp (x). Under controlled conditions, a pilot plant manufactures 16 samples, each from differential batch of pulp, and measures the tensile strength. Determine if there is significance relationship between x and y.
y = 101, 117, 117, 106, 132, 147, 147, 134, 111, 123, 125, 145, 134, 145, 144, 146.9
x = 1.0, 1.5, 1.5, 1.5, 2.0, 2.0, 2.2, 2.4, 2.5, 2.5, 2.8, 2.8, 3.0, 3.0, 3.2, 3.3
Mathematics
1 answer:
Elena-2011 [213]2 years ago
3 0

Answer:

At a significance level of 0.05, there is enough evidence to claim that there is a significant relationship between x and y.

P-value = 0.003.

Step-by-step explanation:

If we perform a regression analysis relating x and y, we get the best fitting line with equation:

y=15.82x+92.9

and a correlation coefficient r:

r=0.693

We have to test the hypothesis, where the alternative hypothesis claims that there is a relationship between these two variables, and the null hypothesis claiming there is no relationship (meaning that the correlation is not significantly different from 0).

This can be written as:

H_0: \rho=0\\\\H_a:\rho\neq0

where ρ is the population correlation coefficient for x and y.

The significance level is assumed to be 0.05.

The sample size is n=16.

The degrees of freedom are df=14.

df=n-2=16-2=14

The test statistic can be calculated as:

t=\dfrac{r\sqrt{n-2}}{\sqrt{1-r^2}}=\dfrac{0.693\sqrt{14}}{\sqrt{1-(0.693)^2}}=\dfrac{2.593}{0.721}=3.597

For a test statistic t=2.05 and 14 degrees of freedom, the P-value is calculated as:

\text{P-value}=2\cdot P(t>3.597)=0.003

The P-value (0.003) is smaller than the significance level (0.05), so the effect is significant enough.

The null hypothesis is rejected.

At a significance level of 0.05, there is enough evidence to claim that there is a significant relationship between x and y.

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