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Question 1 of 9

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

Data is assumed normally distributed, and parameter estimates for the two blood pressure measurements are:

$({\bar x_S};{s_S}) = (139.21;4.58)\quad \quad \quad \quad \quad \quad ({\bar x_D};{s_D}) = (91.43;3.61)\quad \quad$

What is the $95\%$ confidence interval for the mean systolic pressure?

Question 2 of 9

In a sports study one wants to investigate whether there is a difference in energy consumption for various types of training. We have (for a single person) measured the energy consumed in 10 jogs of 30 minutes and 10 bike rides of 30 minutes. Each jog and ride was on different days. Measurements, expressed in kcal, is given in the table below:

Jogs Bike rides
314 294
340 317
331 317
333 310
329 327
322 300
332 293
330 321
338 307
325 304

The following R code was run:

x1 <- c(314, 340, 331, 333, 329, 322, 332, 330, 338, 325)
x2 <- c(294, 317, 317, 310, 327, 300, 293, 321, 307, 304)
var(x1)
var(x2)
t.test(x1,x2)
t.test(x1,x2, pair = TRUE, mu = 20)


with the following results:

> var(x1)
[1] 57.82222
> var(x2)
[1] 132
> t.test(x1,x2)

Welch Two Sample t-test

data:  x1 and x2
t = 4.6823, df = 15.615, p-value = 0.0002658
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
11.14535 29.65465
sample estimates:
mean of x mean of y
329.4     309.0

> t.test(x1,x2, pair = TRUE, mu = 20)

Paired t-test

data:  x1 and x2
t = 0.1209, df = 9, p-value = 0.9064
alternative hypothesis: true difference in means is not equal to 20
95 percent confidence interval:
12.91852 27.88148
sample estimates:
mean of the differences
20.4


What is the most correct answer to the question: Is there a difference in mean energy consumption between the two types of activities? (Both conclusion and argument should be correct)

Question 3 of 9

We have the following observations of $x_1$, $x_2$ and $y$ on 15 persons:

Person x1 x2 y
1 7.90 16.70 59.00
2 4.60 13.80 44.00
3 5.10 20.20 59.00
4 5.50 14.20 48.00
5 5.20 12.80 45.00
6 6.50 18.60 59.00
7 4.90 20.80 57.00
8 4.60 15.20 45.00
9 4.80 20.50 59.00
10 4.50 22.90 61.00
11 3.80 15.70 46.00
12 4.20 12.30 40.00
13 5.40 16.80 49.00
14 5.80 14.60 47.00
15 4.20 20.50 57.00

And the following R code was run:

myfit <- lm(y~x1 + x2)
summary(myfit)


with the following results:

> summary(myfit)

Call:
lm(formula = y ~ x1 + x2)

Residuals:
Min      1Q  Median      3Q     Max
-2.9092 -1.0104  0.5670  0.9823  1.5360

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)   4.1793     2.8289   1.477    0.165
x1            2.6886     0.3736   7.196 1.09e-05 ***
x2            1.9769     0.1155  17.113 8.54e-10 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 1.438 on 12 degrees of freedom
Multiple R-squared:  0.9654,	Adjusted R-squared:  0.9597
F-statistic: 167.5 on 2 and 12 DF,  p-value: 1.709e-09


What kind of analysis is done here?

Question 4 of 9

We repeat from the question above:

We have the following observations of $x_1$, $x_2$ and $y$ on 15 persons:

Person x1 x2 y
1 7.90 16.70 59.00
2 4.60 13.80 44.00
3 5.10 20.20 59.00
4 5.50 14.20 48.00
5 5.20 12.80 45.00
6 6.50 18.60 59.00
7 4.90 20.80 57.00
8 4.60 15.20 45.00
9 4.80 20.50 59.00
10 4.50 22.90 61.00
11 3.80 15.70 46.00
12 4.20 12.30 40.00
13 5.40 16.80 49.00
14 5.80 14.60 47.00
15 4.20 20.50 57.00

And the following R code was run:

myfit <- lm(y~x1 + x2)
summary(myfit)


with the following results:

> summary(myfit)

Call:
lm(formula = y ~ x1 + x2)

Residuals:
Min      1Q  Median      3Q     Max
-2.9092 -1.0104  0.5670  0.9823  1.5360

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)   4.1793     2.8289   1.477    0.165
x1            2.6886     0.3736   7.196 1.09e-05 ***
x2            1.9769     0.1155  17.113 8.54e-10 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 1.438 on 12 degrees of freedom
Multiple R-squared:  0.9654,	Adjusted R-squared:  0.9597
F-statistic: 167.5 on 2 and 12 DF,  p-value: 1.709e-09


What is the only correct statement among the following to make here?

Question 5 of 9

Use the situation described in the exercise above, repeated here again:

> summary(myfit)

Call:
lm(formula = y ~ x1 + x2)

Residuals:
Min      1Q  Median      3Q     Max
-2.9092 -1.0104  0.5670  0.9823  1.5360

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)   4.1793     2.8289   1.477    0.165
x1            2.6886     0.3736   7.196 1.09e-05 ***
x2            1.9769     0.1155  17.113 8.54e-10 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 1.438 on 12 degrees of freedom
Multiple R-squared:  0.9654,	Adjusted R-squared:  0.9597
F-statistic: 167.5 on 2 and 12 DF,  p-value: 1.709e-09


What is the estimate of the residual standard deviation, $\hat{\sigma}$?

Question 6 of 9

Use again the situation described in the exercise above and repeated here again again:

> summary(myfit)

Call:
lm(formula = y ~ x1 + x2)

Residuals:
Min      1Q  Median      3Q     Max
-2.9092 -1.0104  0.5670  0.9823  1.5360

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)   4.1793     2.8289   1.477    0.165
x1            2.6886     0.3736   7.196 1.09e-05 ***
x2            1.9769     0.1155  17.113 8.54e-10 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 1.438 on 12 degrees of freedom
Multiple R-squared:  0.9654,	Adjusted R-squared:  0.9597
F-statistic: 167.5 on 2 and 12 DF,  p-value: 1.709e-09


What is the $95\%$ confidence interval for $\beta_1$ the relation between $x_1$ and $y$?

Question 7 of 9

Ten students took a mathematics test with 25 questions with the following results (number of correct answers): 9, 18, 19, 21, 25, 25, 21, 19, 16, 7.

Which one of the following statements is true? (use the definition from Chapter 1)

Question 8 of 9

Ten students took a mathematics test with 25 questions with the following results (number of correct answers): 9, 18, 19, 21, 25, 25, 21, 19, 16, 7.

What is the sample variance $s^2$ for these numbers?

Question 9 of 9

When making statistical hypothesis tests we often assume that the significance level $\alpha$ is $5\%$.

This means that: