IB Maths AI HL · Paper 3

The graph shows one full period over 669 days, so the period is 669 days.

Mars turns through $2\pi$ radians in $24\frac{2}{3}$ hours.

The angle $\omega$ is largest when $\cos \omega$ is smallest, so take $\delta =-0.440$.

The smallest $\omega$ comes from the other end, $\delta =0.440$.

The time until sunrise is the angle divided by the rotation rate per hour.

Do the same with the smallest angle.

The amplitude is half the difference between the maximum and minimum.

The midline is halfway between the maximum and minimum.

The day length is sunset minus sunrise: $L=S-R$. The constants combine as $18.65-c$.

Write each sine as the imaginary part of a complex exponential with constant modulus. Then $f(t)=\mathrm{Im}(z_1-z_2)$ with

The shortest daylight is when the sine takes its lowest value, $-1$.

A stratified sample takes a share in proportion to each department's size.

The scatter diagram shows two groups. The managers in the two departments seem to score differently: the international manager appears to be less generous. Pooling both would mix two different marking scales, so a single correlation coefficient would not be meaningful.

A strong positive value means the written assessment marks agree closely with the managers' scores, so the assessment is likely to be a valid measure of performance in this department.

Giving the same test twice to the same people is a test–retest check of reliability.

A strong positive correlation between the two sittings means the assessment gives consistent results, so it appears reliable.

Each of the 5 sections is compared with each of the 5 attributes.

If there is no correlation, each test has a 0.05 chance of a false "significant" result. The chance of none among 25 independent tests is $0.95^{25}$.

There is evidence of a correlation between section 2 and attribute X. But with 25 tests, a significant result is expected to turn up by chance about once, so it should be treated cautiously and checked with more data.