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	90	177	181	354	370	442	963
y	0.39	0.60	0.52	0.66	0.61	0.69	0.92

(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 xincreases in this way, is there substantial evidence that the true average increase in y is less than 0.6? (Use α = 0.05.)

H₀: β₁ = 0.0006667
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	=

(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.)

(INTERVAL, FORM) mg/m^3

Answer 1

b)

The regression analysis is done in excel by following steps;

Step 1: Write the data values in excel. The screenshot is shown below,

Step 2: DATA > Data Analysis > Regression > OK.The screenshot is shown below,

Step 3: Select Input Y Range: 'y' column, Input X Range: 'x' column then OK. The screenshot is shown below,

The result is obtained. The screenshot is shown below,

The regression equation is,

The R square value is,

Which means Independent variable 'x' explains the 89.57% of the variance of Dependent Variable.

c)

The t value is obtained using the formula,

The P-value is obtained using the excel function =T.DIST(x,deg_freedom) where x is the t value degree of freedom = n-1=7-1=6.

The P-value = 0.07124 is lgreater than than 0.05 at 5% significance level. The null hypothesis cannot be rejected. Now we can state that there is a no significant evidence that true average is less than 0.000667

d)

The regression equation is,

For change in 'x' = 1 cm/sec,

The 95% confidence interval is obtained by using the formula,