Topic 1: Integration
TECH ACTIVE
Integrate 8x^2 from 1 < x < 4
168
What two things need to be true for a function to be a probability density function?
- All values of f(x) are >= 0
- The total area under the probability density function = 1
Give an example of bias from the following:
Asking only students who failed the last Maths exam which topics should be reviewed before the next exam.
Selection bias: Only students who failed are being surveyed, so the results may not represent the needs of the entire class.
These students may focus on topics they personally struggled with, even if other topics would benefit the majority of students
TECH ACTIVE
Determine the area between x^4 - 8 and the x-axis.
22.63 units^2
TECH ACTIVE
For a standard normal variable, determine P(Z>1.5), and draw an appropriate diagram.
0.0668
What would be an appropriate way to collect data on whether students find Monday morning Physics revision sessions beneficial?
Conduct an anonymous survey of students who attend the sessions, using neutral questions about their usefulness.
Surveying a representative range of students and allowing anonymous responses can help reduce bias
TECH FREE
Integrate 6e^-x from 0 to 4
6(1 - e^-4)
TECH ACTIVE
James scored 92% on his Maths exam. If the cohort mean was 85% and the standard deviation was 4%, James performed in the top what percentage?
Top 4% of the cohort (approximately 4.01%).
TECH FREE
72 out of 120 people surveyed were happy that Australia won the 2026 FIFA World Cup.
Determine the mean and standard deviation of the sample proportion.
Mean = 0.6
S.D. = 0.0447
TECH FREE
Determine the area between x^3 and the x-axis, from -2 < x < 2
8
TECH FREE
A probability density function is given by:
ax for 0<x<3
a) Determine the value of a.
b) Determine the expected value of the p.d.f.
a) 2/9
b) 2
James rolls a fair six-sided die five times. He rolls a 6 on all five occasions and assumes the die is biased.
Is he correct? Explain your answer.
Not necessarily. Although rolling five consecutive sixes is very unlikely, it is still possible with a fair die.
A larger number of trials would be needed to gather stronger evidence that the die is biased
TECH FREE
Determine the area between the functions x^2 and 3x + 10
43 1/6
TECH ACTIVE
The weight of oranges in a particular batch is normally distributed.
One orange is 250 g and is 1 standard deviation above the mean. Another orange is 220 g and is 1/2 standard deviation below the mean.
Determine the mean and standard deviation of the oranges
S.D. = 20
Mean = 230
TECH ACTIVE
60% of the population has watched 'Harry Potter'. If a sample of 20 people was chosen, determine the probability that the sample proportion matches the population proportion.
0.1797