02323 · Test Quiz 12
Question 1 of 6
The polar moment of inertia of a tube is calculated by the formula:
\({I_P} = \frac{\pi }{ {32}} \cdot \left( { {D^4} - {d^4}} \right)\)
where D is the outer diameter of the tube and d is the inner diameter of the tube
We assume that $\left( { {\mu _D};{\sigma _D}} \right) = (45;0.3\,)\,\text{mm}\quad \mbox{and}\quad \left( { {\mu _d};{\sigma _d}} \right) = (32;0.4{\kern 1pt} )\,\text{mm}$ and that the two diameters are independent.
The probability that the diameter difference (D-d) at a given location is less than 12.5 mm is:
Question 2 of 6
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. 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 |
Consider only data for the bike rides. What are respectively the lower quartile, median and upper quartile for these data? (It is assumed that the textbook definitions are used. These differ slightly from R’s definitions.)
Question 3 of 6
Let $ X_ {1} $ describe the weight of a random cola bottle filled with 0.25 l. Let $ X_ {2} $ describe the weight of a random empty cola bottle. It is assumed that these variables are normally distributed and independent with mean and variance as given here:
\[{X_1}:\quad \;({\mu _{ {X_1}}};\sigma _{ {X_1}}^2) = (626.9 g;4.{2^2} {g^2})\;\quad \mbox{and} \quad {X_2}:\quad ({\mu _{ {X_2}}};\sigma _{ {X_2}}^2) = (364.7 g;3.{6^2} {g^2})\;\]The probability that the weight of a random filled bottle of cola exceeds the weight of a random empty bottle with more than 265 g is:
Question 4 of 6
If you did the previous exercise, the following is a repetition: 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. 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=T,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=T,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
A new study in energy consumption during by bike rides is planned. We would like a 95\% confidence interval for $ \mu $ with a total width of $ 8$ kcal. How many bike rides should we approximately carry out to achieve this precision?
Question 5 of 6
A strength calculation on an old tube in a construction is to be performed. Because of corrosion and age diameters are quite ‘indeterminate’. Therefore several measurements are made of as well outer as inner diameter. The measurements of outer respectively inner diameter are independent of each other. The results are listed below: (all dimensions in mm)
Outer diameter, x: 44.9, 44.2 , 44.6, 44.8 , 44.0, 45.1
Inner diameter, y: 32.4, 32.5, 31.5, 32.2, 32.6, 31.7
\[\left( {\bar x;{s_x}} \right) = (44.6;0.424)\,\mathrm{mm}\quad \mbox{and} \quad \left({\bar y;{s_y}} \right) = (32.15;0.451)\,\mathrm{mm}\]Assume that the standard deviation of the measurements on the outer diameter is $\sigma_x = 0.43$. A new study is planned in which the outer diameter should be determined with an accuracy corresponding to a 95% confidence interval of size $\pm $ 0.1.
What number of measurements, $ n $ is required?
Question 6 of 6
A used car dealer sells a car to a buyer, even though the dealer knows that the car will have a major breakdown within the next 6 month. The dealer provides a warranty of 1.5 months. Let X represent the length of time until the breakdown occurs. Assume that X is a uniform random variable with values between 0 and 6 months.
The probability that the breakdown occurs while the car is still under warranty is approximately: