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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 substantially lowered the workplace standard. An article gave the accompanying data on [tex]x = \text{fluid-flow velocity for a } 5\% \text{ soluble oil } ( \text{cm/sec} )[/tex] and [tex]y = \text{the extent of mist droplets having diameters smaller than } 10 \mu m \left( \text{mg/m}^3 \right)[/tex]:

\begin{tabular}{c|ccccccc}
[tex]$x$[/tex] & 87 & 177 & 190 & 354 & 367 & 442 & 963 \\
\hline
[tex]$y$[/tex] & 0.43 & 0.60 & 0.49 & 0.66 & 0.62 & 0.69 & 0.96
\end{tabular}

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

A. Yes, a scatter plot shows a reasonable linear relationship.
B. 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.)

[tex]\boxed{}[/tex]

(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 [tex]x[/tex] values in the sample). When [tex]x[/tex] increases in this way, is there substantial evidence that the true average increase in [tex]y[/tex] is less than 0.6? (Use [tex]\alpha = 0.05[/tex].)

State the appropriate null and alternative hypotheses.

[tex]
\begin{aligned}
H_0: \beta_1 = 0.0006667 \\
H_a: \beta_1 \neq 0.0006667
\end{aligned}
[/tex]

[tex]
\begin{array}{l}
H_0: \beta_1 = 0.0006667 \\
H_a: \beta_1 \ \textgreater \ 0.0006667 \\
H_0: \beta_1 \neq 0.0006667 \\
H_a: \beta_1 = 0.0006667 \\
H_0: \beta_1 = 0.0006667
\end{array}
[/tex]

Sagot :

GinnyAnsweridn
Let's address each of the questions step by step:

### Part (a)
The question asks whether a scatter plot of the data supports the strategy of performing a simple linear regression analysis to relate fluid-flow velocity ([tex]\(x\)[/tex]) and the extent of mist droplets ([tex]\(y\)[/tex]).

- Answer:
Yes, a scatter plot shows a reasonable linear relationship.

### Part (b)
We need to determine the proportion of observed variation in mist that can be attributed to the simple linear regression relationship between velocity and mist.

- Answer:
The proportion of observed variation in mist explained by the regression model is given by [tex]\(R^2\)[/tex], which in this case is [tex]\(0.937\)[/tex] (rounded to three decimal places).

### Part (c)
The question asks whether there is substantial evidence that the true average increase in [tex]\(y\)[/tex] is less than 0.6 when [tex]\(x\)[/tex] increases from 100 to 1000, using a significance level of [tex]\(\alpha=0.05\)[/tex].

1. State the appropriate null and alternative hypotheses:

- Correct hypotheses:
[tex]\[ \begin{aligned} H_0: & \ \beta_1 = 0.0006667 \\ H_a: & \ \beta_1 < 0.0006667 \end{aligned} \][/tex]

2. Perform the hypothesis test:

- The test statistic ([tex]\(t_{stat}\)[/tex]) value is found to be [tex]\(-1.517\)[/tex]. This compares the estimated slope to the null hypothesis slope.

- The critical value for a one-tailed t-test at [tex]\(\alpha = 0.05\)[/tex] with [tex]\(n-2\)[/tex] degrees of freedom ([tex]\(df = 5\)[/tex]) is approximately [tex]\(2.015\)[/tex].

3. Make the decision:

- Since [tex]\(t_{stat} = -1.517\)[/tex] is not less than [tex]\(-2.015\)[/tex], we do not reject the null hypothesis.

- Conclusion:
There is not enough evidence at the [tex]\(\alpha=0.05\)[/tex] significance level to conclude that the true average increase in [tex]\(y\)[/tex] is less than 0.6 when [tex]\(x\)[/tex] increases from 100 to 1000.

To summarize:
(a) Yes, a scatter plot shows a reasonable linear relationship.
(b) The proportion of observed variation in mist explained by the regression model is 0.937.
(c) The hypotheses are:
[tex]\[ \begin{aligned} H_0: & \ \beta_1 = 0.0006667 \\ H_a: & \ \beta_1 < 0.0006667 \end{aligned} \][/tex]

There is not enough evidence at [tex]\(\alpha = 0.05\)[/tex] to reject the null hypothesis.
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