02402 · Test Quiz 8

For oil nozzles, it is assumed that the nozzle capacity at a given atomizing pressure can be written as: \(Q_2=Q_1\cdot \sqrt{\frac{P_2}{P_1}},\) where $ Q_1 $ is the reference capacity at the pressure $ P_1 $ and $ Q_2 $ is the capacity at the pressure $ P_2 $. To verify the formula, a study was performed where the capacity at 14 bar is investigated as a function of the capacity at 10 bar (reference pressure). The following values are found:

The following computations can be used ($x=Q_1$ and $y=Q_2$) \(\bar{x}=2.635, \bar{y}=3.115, S_{xx}=7.5655, S_{yy}=10.499, S_{xy}=8.9090\) The following R-code was run:

	Q1=c(1.46,1.66,1.87,2.11,2.37,2.67,2.94,3.31,3.72,4.24)
	Q2=c(1.76,1.96,2.24,2.46,2.80,3.11,3.46,3.95,4.42,4.99)
	summary(lm(Q2~Q1))

It gave the following output:

Call:
lm(formula = Q2 ~ Q1)

Residuals:
  	Min        1Q    Median        3Q       Max 
-0.046215 -0.014807 -0.004898  0.026954  0.040131 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  0.01207    0.03153   0.383    0.712    
Q1           1.17758    0.01136 103.622  8.4e-14 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 

Residual standard error: 0.03126 on 8 degrees of freedom
Multiple R-squared: 0.9993,     Adjusted R-squared: 0.9992 
F-statistic: 1.074e+04 on 1 and 8 DF,  p-value: 8.403e-14

What is the proportion of $y$-variability explained and the estimated (residual) standard deviation ($s_e$)?

If you did the previous exercise, the following is a repetition:

For oil nozzles, it is assumed that the nozzle capacity at a given atomizing pressure can be written as: \(Q_2=Q_1\cdot \sqrt{\frac{P_2}{P_1}},\) where $ Q_1 $ is the reference capacity at the pressure $ P_1 $ and $ Q_2 $ is the capacity at the pressure $ P_2 $. To verify the formula, a study was performed where the capacity at 14 bar is investigated as a function of the capacity at 10 bar (reference pressure). The following values are found:

The following computations can be used ($x=Q_1$ and $y=Q_2$) \(\bar{x}=2.635, \bar{y}=3.115, S_{xx}=7.5655, S_{yy}=10.499, S_{xy}=8.9090\) The following R-code was run:

	Q1=c(1.46,1.66,1.87,2.11,2.37,2.67,2.94,3.31,3.72,4.24)
	Q2=c(1.76,1.96,2.24,2.46,2.80,3.11,3.46,3.95,4.42,4.99)
	summary(lm(Q2~Q1))

It gave the following output:

Call:
lm(formula = Q2 ~ Q1)

Residuals:
  	Min        1Q    Median        3Q       Max 
-0.046215 -0.014807 -0.004898  0.026954  0.040131 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  0.01207    0.03153   0.383    0.712    
Q1           1.17758    0.01136 103.622  8.4e-14 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 

Residual standard error: 0.03126 on 8 degrees of freedom
Multiple R-squared: 0.9993,     Adjusted R-squared: 0.9992 
F-statistic: 1.074e+04 on 1 and 8 DF,  p-value: 8.403e-14

Is data in correspondence with the hypothesis corresponding to the assumption above that the slope should be $\sqrt{14/10}=1.1832$? (As well answer as argument must be valid)

If you did the previous exercise, the following is a repetition:

For oil nozzles, it is assumed that the nozzle capacity at a given atomizing pressure can be written as: \(Q_2=Q_1\cdot \sqrt{\frac{P_2}{P_1}},\) where $ Q_1 $ is the reference capacity at the pressure $ P_1 $ and $ Q_2 $ is the capacity at the pressure $ P_2 $. To verify the formula, a study was performed where the capacity at 14 bar is investigated as a function of the capacity at 10 bar (reference pressure). The following values are found:

The following computations can be used ($x=Q_1$ and $y=Q_2$) \(\bar{x}=2.635, \bar{y}=3.115, S_{xx}=7.5655, S_{yy}=10.499, S_{xy}=8.9090\) The following R-code was run:

	Q1=c(1.46,1.66,1.87,2.11,2.37,2.67,2.94,3.31,3.72,4.24)
	Q2=c(1.76,1.96,2.24,2.46,2.80,3.11,3.46,3.95,4.42,4.99)
	summary(lm(Q2~Q1))

It gave the following output:

Call:
lm(formula = Q2 ~ Q1)

Residuals:
  	Min        1Q    Median        3Q       Max 
-0.046215 -0.014807 -0.004898  0.026954  0.040131 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  0.01207    0.03153   0.383    0.712    
Q1           1.17758    0.01136 103.622  8.4e-14 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 

Residual standard error: 0.03126 on 8 degrees of freedom
Multiple R-squared: 0.9993,     Adjusted R-squared: 0.9992 
F-statistic: 1.074e+04 on 1 and 8 DF,  p-value: 8.403e-14

For a random nozzle with capacity 4.45 kg/h at 10 bar, the capacity is measured at 14 bar. Within which limits is this measurement expected to be with 99% confidence:

For a device for measuring blood pressure at home the accuracy was investigated. Therefore repeated measurements of blood pressure of a person, with a time interval of 5 min and under as identical circumstances as possible. The following data were measured:

The following bootstrap procedure was performed in R for the measurements of the systolic blood pressure:

x=c(143,134,138,138,135,131,135,139,141,143,142,141,149,140)
k=10000
mybootstrapsamples=replicate(k,sample(x,replace=TRUE))
mybootstrapmeans=apply(mysamples,2,mean)

And eight different percentiles of the bootstrap distribution was:

>quantile(mybootstrapmeans,c(0.005,0.01,0.025,0.05,0.95,0.975,0.99,0.995))
   	0.5%       1%     2.5%       5%      95%    97.5%      99%    99.5% 
136.2143 136.5000 136.9286 137.2857 141.2143 141.5714 141.9286 142.2143

A 99% confidence interval for the mean value of the systolic blood pressure becomes: (based on the displayed bootstrap distribution)