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Question 1 of 4

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:

 Capacity $Q_1$ (kg/h) 1.46 1.66 1.87 2.11 2.37 2.67 2.94 3.31 3.72 4.24 at the pressure 10 bar Capacity $Q_2$ (kg/h) 1.76 1.96 2.24 2.46 2.80 3.11 3.46 3.95 4.42 4.99 at the pressure 14 bar

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 ($s_e$)?

Question 2 of 4

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:

 Capacity $Q_1$ (kg/h) 1.46 1.66 1.87 2.11 2.37 2.67 2.94 3.31 3.72 4.24 at the pressure 10 bar Capacity $Q_2$ (kg/h) 1.76 1.96 2.24 2.46 2.80 3.11 3.46 3.95 4.42 4.99 at the pressure 14 bar

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)

Question 3 of 4

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:

 Capacity $Q_1$ (kg/h) 1.46 1.66 1.87 2.11 2.37 2.67 2.94 3.31 3.72 4.24 at the pressure 10 bar Capacity $Q_2$ (kg/h) 1.76 1.96 2.24 2.46 2.80 3.11 3.46 3.95 4.42 4.99 at the pressure 14 bar

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:

Question 4 of 4

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:

Measurement no 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Systolic pressure (mmHg) 143 134 138 138 135 131 135 139 141 143 142 141 149 140
Diastolic pressure (mmHg) 98 94 96 89 88 95 85 88 89 92 89 92 93 92

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)