Question

Mist (airborne droplets or aerosols) is generated when metal-removing fluids are used in machining operations to cool and lubricate the tool and workpiece. Mist generation is a concern to OSHA, which has recently lowered substantially the workplace standard. An article gave the accompanying data on x = fluid-flow velocity for a 5% soluble oil (cm/sec) and y = the extent of mist droplets having diameters smaller than 10 µm (mg/m³):

x	88	177	187	354	360	442	968
y	0.38	0.60	0.49	0.66	0.61	0.69	0.99

(a) The investigators performed a simple linear regression analysis to relate the two variables. Does a scatter plot of the data support this strategy?

Yes, a scatter plot shows a reasonable linear relationship.No, a scatter plot does not show a reasonable linear relationship.    

(b) What proportion of observed variation in mist can be attributed to the simple linear regression relationship between velocity and mist? (Round your answer to three decimal places.)


(c) The investigators were particularly interested in the impact on mist of increasing velocity from 100 to 1000 (a factor of 10 corresponding to the difference between the smallest and largest x values in the sample). When x increases in this way, is there substantial evidence that the true average increase in y is less than 0.6? (Use α = 0.05.)
State the appropriate null and alternative hypotheses.

A) H₀: β₁ = 0.0006667
B) H_a: β₁ ≠ 0.0006667H₀: β₁ = 0.0006667
C) H_a: β₁ > 0.0006667    H₀: β₁ ≠ 0.0006667
D) H_a: β₁ = 0.0006667H₀: β₁ = 0.0006667
E) H_a: β₁ < 0.0006667

Calculate the test statistic and determine the P-value. (Round your test statistic to two decimal places and your P-value to three decimal places.)

t	=
P-value	=

State the conclusion in the problem context.

Fail to reject H₀. There is not sufficient evidence that with an increase from 100 to 1000, the true average increase in y is less than 0.6.Reject H₀. There is not sufficient evidence that with an increase from 100 to 1000, the true average increase in y is less than 0.6.    Reject H₀. There is sufficient evidence that with an increase from 100 to 1000, the true average increase in y is less than 0.6.Fail to reject H₀. There is sufficient evidence that with an increase from 100 to 1000, the true average increase in y is less than 0.6.

(d) Estimate the true average change in mist associated with a 1 cm/sec increase in velocity, and do so in a way that conveys information about precision and reliability. (Calculate a 95% CI. Round your answers to six decimal places.)
(____,_____) mg/m³

Answer 1