Question

Mist (airborne droplets or aerosols) is generated when metal-removing fluids are used in machining operations to cool and lubricate the tool and work-piece. Mist generation is a concern to OSHA, which has recently lowered substantially the workplace standard. The article "Variables Affecting Mist Generation from Metal Removal Fluids" (Lubrication Engr., 2002: 10-17) 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 some value:

x: 89 177 189 354 362 442 965

y: .40 .60 .48 .66 .61 .69 .99

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. Hint: This question is asking for a CI for beta. Compute it AND interpret it. By hand; i.e. you must use the basic formulas for the CI. E.g. for beta: beta_hat +- t* s_e/sqrt(S_xx) , but you may use R to compute the various terms in the formula. Use 95% confidence level.

e. Suppose the fluid velocity is 250 cm/sec. Compute an interval estimate of the corresponding mean y value. Use 95% confidence level. Interpret the resulting interval. By hand, as in part d.

f. Suppose the fluid velocity for a specific fluid is 250 cm/sec. Predict the y for that specific fluid in a way that conveys information about precision and reliability. Use 95% prediction level. Interpret the resulting interval. By hand, as in part d.

Answer 1

The regression equation is
y = 0.4041 + 0.00062 x

Predictor	Coef	SE Coef	T	P
Constant	0.40412	0.03459	11.68	0.000
x	0.00062	0.00007579	8.19	0.000

S = 0.0540453 R-Sq = 93.1% R-Sq(adj) = 91.7%

x	y	(x-X_)^2
89	0.4	32144.9041
177	0.6	8333.8641
189	0.48	6286.9041
354	0.66	7346.2041
362	0.61	8781.5641
442	0.69	30175.1641
965	0.99	485404.8241
	Sxx=	578473.4287