Mock Exam Set 1 - Paper 3

[Maximum mark: 27]

Every year a sports academy enrols new members from a large group of applicants. This year the academy enrols $100$ new members from whom the academy records some data and conducts several tests.

A staff member wonders if there exists some relationship between the age and the EQ (emotional intelligence) of new members. The data recorded is shown in the following table.

1. A $\chi^2$ test is performed at the 5% significant level to determine whether the EQ test result is independent of age.

   1. State the null and alternative hypotheses for this test.

   2. Determine the number of degrees of freedom for this test.

   3. Find the $p$-value for this test.

   4. Write down the conclusion to the test. [6]
      

From the total of 100 new members enrolled, 34 selected tennis, and 66 selected volleyball as their sport of choice. A sample of 30 new members is selected to participate in a sports competition. The sample will contain the same proportion of tennis and volleyball players as the population of 100 new members.

2. Show that $10$ tennis members were selected for the competition.[2]
   

3. The selected members that play tennis each stayed in a different hotel during the competition. The following table shows the star rating of each hotel and the average guest satisfaction scored on a 100 point scale.

   

   1. Find the Pearson's product moment correlation coefficient, $r$.

   2. Interpret the value found in part (i).

   3. Find the Spearman's rank correlation coefficient, $r_s$.

   4. Interpret the value found in part (iii).

   5. In the context of this question, comment if both correlation coefficients should be considered when determining the relationships between guest satisfaction and the star rating of hotels or if one should be used instead of the other.[9]
      

The selected tennis group is tested on motor skills twice, at the beginning and after the tournament to measure its reliability. Results are shown in the following table.

4. 1. State the name of this type of test for reliability.

   2. Find the Pearson's product moment correlation coefficient, $r$.

   3. Interpret the value found in part (ii).

   4. Hence, comment on the reliability of the motor abilities test.

   5. Determine at the 5% level whether the test indicates that the selected tennis group improved their motor skills during the tournament. [10]
      

[Maximum mark: 28]

Annika, Bob, Chloe and Dani go to a fair and take a ride on a Ferris wheel. The wheel has $8$ evenly spaced passenger cars as shown in the following diagram. The wheel completes one rotation in the anticlockwise direction every 80 seconds.

Annika starts her ride at position A. Let $h_A$ be Annika's height in meters after $t$ seconds. $h_\mathrm{A}$ can be modelled by the function

The diagram below shows the graph of $y=h_A(t)$ for one revolution.

1. Show that

   1. $a=14$.

   2. $b=\dfrac{\pi}{40}$.

   3. $d=18$. [3]
      

At the time Annika starts her ride, Bob is at position B. Bob's position can be modeled by the function $h_\mathrm{B}(t)=14\cos\Big(\dfrac{\pi}{40}t\Big)+18$.

2. Determine

   1. the first time at which Annika and Bob are at the same height.

   2. the height at which this occurs. [4]
      

At the time Annika starts her ride, Chloe is at position C. Chloe's position at time $t$ can be written as $h_\mathrm{C}(t)=m(h_\mathrm{A}(t))+n$.

3. Find the value of $m$ and the value of $n$. [3]
   

When Chloe reaches an altitude of $30$ meters she has a view of the whole town.

4. 1. Find the time when Chloe has a view of the whole town for the first time.

   2. Find the angle rotated through by Chloe's car from its start position to the point when she has a view of the whole town for the first time. Give your answer in radians, correct to one decimal place. [4]
      

Dani is at position D when Annika starts her ride.

5. Find the value of $c$ such that the function $h_\mathrm{D}(t)=14\sin\Big(\dfrac{\pi}{40}(t-c)\Big)+18$ describes Dani's height at time $t$. [3]
   

The function $D(t)$ represents the difference in height between Annika and Dani's cars.

6. Write $D(t)$ as the difference of two sine functions. [2]
   

$D(t)$ can be written in the form $\text{Im}(z_1-z_2)$, where $z_1$ and $z_2$ are complex functions of $t$.

7. 1. Write $z_1$ and $z_2$ in exponential form.

   2. Hence or otherwise find an equation for $D(t)$ in the form $D(t)=p\sin(qt+r)+s,$ where $p,q,r,s\in \mathbb{R}$.

   3. Find the maximum difference in height between Annika and Dani's cars. [9]