The number of daily visitors to the famous 'r3 u/z-xfpx:/ akv7zsBondi Beach' in Sydney, Australia, and the daily temperature on those days, were recorded for eig3 k/:sxuxa7zrfpz-v /ht days in February. The table below shows this data and has been ordered in ascending order by temperature.


The number of visitors to Bondi Beach varies linearly with the temperature.
1. Find
1. Pearson's product-moment correlation coefficient, r ;≈   
2. the equation of the regression line y on x . y =ax+b a =    b =   
2. Use the equation of the regression line y on x to estimate the number of visitors to Bondi Beach during a day the temperature is $26^{\circ} \mathrm{C}$ .

  At Changi Airport in Singapore, six passengtq7b aocgm9uz/q, 2tdo1(h8t ers were surveyed on the number of stores they had shopped in, and the waiting time for the flight, in minutes. The daozhb(qdqmg/a uo271t , t c8t9ta is shown in the scatter diagram below, where x represents the waiting time for their flight and y represents the number of stores they had shopped in.

1.Use the scatter diagram above to complete the missing cells in the table below.


2. Find the Pearson's product moment correlation coefficient, r , for this data.≈   
3. Give two reasons why it would not be valid to use this scatter diagram to predict the number of stores a passenger will shop in, who has a 120 minute wait time.

  The following table shows the total revenue, y , in Ause67 u x5jhe9 linec9t1tralian dollars (AUD), obtained daily during the first week of January 2020 ,j19in 9eth 67clxe5eu by Peppy's Pizza restaurant and the number of guests, x , served.



1. 1. Calculate the Pearson's product-moment correlation coefficient, r , for this data.   
2. Hence comment on the result.
2. Write down the equation of the regression line y on x .y =ax+b a =    b =   
3. Use the equation of the regression line to estimate the revenue of serving 70 guests. Give your answer correct to the nearest AUD.≈   

  The average price for one square metre of landn(x*g bg 6x-jtjj3z;v:q cavzj yn;) 3 is recorded for seven different locations in Melbourne, Australia. The following table shows the average prices for these locationcj:ayj6)t j(;vn3bnvq x;-gzzg* j 3 xs and the distances from the centre of the Melbourne city. This data is also shown on the scatter diagram below.



1. Use the scatter diagram to find the value of a and b in the table. a =    b =   
2. Calculate
1. the mean distance from the Melbourne city centre, $\bar{x}$ ; ≈    km
2. the mean 1 $\mathrm{~m}^{2}$ price, in thousands AUD, $\bar{y}$ ; ≈    thousands AUD
3. the Pearson's product-moment correlation coefficient, r . ≈   
3. Write down the equation of the regression line y on x .y =ax+b a =    b =   
4. Use the line of the regression equation to estimate the price of 1 $\mathrm{~m}^{2}$ of the land 4 $\mathrm{~km}$ from the Melbourne city centre. State whether your estimate is reliable and justify your answer.

  A fast food store own5/2a2uap kn 53lqg bohckgfl4p0/o3n er recorded the daily number of burgers and hot dogs sold in her restaurant. The followinug3p5oh/53oqblcn2p 2lfkk 4a0 ag n/g table shows this data for the last eight days. This data is also shown on the scatter diagram below.


The number of burgers sold varies linearly with the number of hot dogs sold.
1. Find
1. the mean number of burgers, $\bar{x}$ ;   
2. the mean number of hot dogs, $\bar{y}$ .   
2. Find the line of regression equation, y on x , and plot on the scatter diagram above.
3. Use the line of regression equation, y on x , to estimate the number of hot dogs sold on a day when 22 burgers are sold.≈   

  A health study was conducted on 8 male participants, measuring theii 2+y,key cen6r average daily sugar intake (in grams) and their bo,26ie ycek+nydy fat percentage. This data were used to determine whether sugar intake influences male body fat levels. The results are shown in the table.



1. Use your graphic calculator to find
1. $\bar{x}$ , the mean sugar intake;   
2. $\bar{y}$ , the mean body fat percentage;   
3. r , the Pearson's product-moment correlation coefficient.≈   

The equation of the regression line y on x is in the form y=m x+c .
2. 1. Find the values of m and c for this data. m =    c =   
2. Show that the point $\mathrm{M}(\bar{x}, \bar{y})$ lies on the regression line y on x .

A ninth male participant, Daniel, has the average daily sugar intake of 32 grams.
3. 1. Use the regression line y on x to estimate Daniel's body fat percentage. ≈    %
2. Justify whether it is valid to use the regression line y on x to estimate Daniel's body fat percentage.

Daniel's body fat percentage is measured to be $12 \%$ .
4. Calculate the percentage error in Daniel's estimated body fat percentage, found in part (c).≈    %

  A clothes tailoring shop received an order for pants of different waiqbx4hb,:jz7 il b/hnb i8 rk1*st and length sizes. The made-to-order measurements for this order were recorded and are shown in thbnr,bz8k4 /ih bqx j* :il7hb1e following table.


1. Find the Pearson's product moment correlation coefficient, r , for this data.≈   
2. Comment on the result found for r .

The equation of the regression line is in the form y=m x+c .
3. 1. Find the value of m and comment on its meaning.
2. Find the value of c and interpret its meaning if appropriate. If not appropriate, explain why.

  Gippsland in Victoria, Australia, is an import-pgxjdk0t z;n( .sz3avj8 3 tiant milk-producing farmland area. Some people might argue that, because of transportation costs, the cost of milk increases with the distance of markets from Gippsland. The table below shows the milk price in eight markets 8ik-jgtp.nz3j( va0 ;sx3dztthe Gippsland farmers supply to.


1. Find the Pearson's product moment correlation coefficient for this data, r .≈   
2. Comment on the result obtained for r .

The equation of the regression line for this data is in the form y=m x+c .
3. 1. Find the value of m and comment on its meaning.
2. Find the value of c and interpret its meaning if appropriate. If not appropriate, explain why.

  The Marketing Director of Sea World was conr; exi n+.i :rqu*47nf8fahoyck o0 /jcerned about the effectiveness of a recent advertising campaign. The Director recorded the advertising cost and the number of tickets sold in the last eight mont*:r.h /koxu84ina qncfrjfy0; ie7+ ohs. The data is shown in the table below and was sorted in ascending order by advertising cost.


Suppose that the number of tickets sold varies linearly with the advertising cost.
1. Find
1. the Pearson's product moment correlation coefficient, r , for this data;≈   
2. the equation of the regression line y on x .y =ax+b a =    b =   
2. The Marketing Director decides to spend 70000 USD on advertisement next month. Use the equation of the regression line found in part (a)(ii) to estimate the number of tickets to the Sea World sold next month. Round your answer to the nearest hundred.

  Wynn conducts an expebv2 3p efd/,hd;tc +mgriment to investigate the rate of photosynthesis, measured by oxygen production (mL), of an algae species over time. Wynn puts six pieces of the algae in six different tubes filled with water and places them at the same distance from the light source. A sample of the oxygen production readings of these tubes after some time are shown in the table b/+3gmd; f,pce b tvdh2elow.


1. 1. Find the line of regression equation for the oxygen production, P , in terms of minutes exposed to the light, t .
2. Interpret your model in the context of this question.
2. 1. Find the coefficient of determination, $r^{2}$ . $r^{2}$ ≈   
2. Interpret the $r^{2}$ in the context of this question.
3. Use the model found in part $(a)$ to predict the oxygen produced after 25 minutes exposed to the light. ≈    mL

  Premier Pet-foods Co. has developed a new dry food for dogs.4yutpozk-5g6 vwz9 8 v They start to mass produce the food and the amount of inventtzo9w kg-ypuv485v z6ory in the warehouse is recorded on a monthly basis. The table below shows the data for the amount of inventory (I, in tonnes) for the first ten months (t).


1.Sketch a scatter plot to represent the given data on the axes below.



2. 1. State a suitable type of function that could model this set of data points.
2. Hence determine the model function for this dataset.$I(t)=at^{2}+b t+c$ a =    b =    c =   
3. Comment on the choice of model using the coefficient of determination.
4. Sketch the model function on the scatter plot and comment on how it fits to the original data.
5 . Using your model, determine after how many months the inventory is at it's maximum.≈    months
6. Using your model, forecast when the the company will run out of inventory.
7. Using your model, determine the amount of inventory after 16 months and comment on if the model is still valid in forecasting inventory levels at this time.

  The table shows the lengty +ru;rjs/s+ vroi)*ah, in km, of an Uber ride in New York City and the corresponding price, in U+j r r)ry*sv+ ;a/siouSD, of the ride.

1. Draw a scatter diagram for the above data on the axes below. Use a scale of 1 unit to represent 2 \mathrm{~km} on the x -axis and 1 unit to represent 5 USD on the u -axis


2. Use your graphic display calculator to find
1. $\bar{x}$ , the mean of the lengths; ≈    km
2. $\bar{y}$ , the mean of the prices. ≈    USD
3. Plot and label the point $\mathrm{M}(\bar{x}, \bar{y})$ on your scatter diagram.
4. Use your graphic display calculator to find
1. the product-moment correlation coefficient, r ;≈   
2. the equation of the regression line y on x .y =ax+b a =    b =   
5 . Draw the regression line y on x on your scatter diagram.
John took a Uber ride of length 22.5 $\mathrm{~km}$ from John F. Kennedy International Airport (JFK) to Central Park.
6. Use the equation of the regression line to estimate the price of John's ride from JFK to Central Park. Give your answer correct to the nearest dollar.≈   
7. Give a reason why it is valid to use your regression line to estimate the price of John's ride.

The actual cost of John's Uber rise was 40 USD.
8. Using your answer to part (f), calculate the percentage error in the estimated price of John's ride.ϵ =    %
​

  At an Apple store in Paris, ix/ms ;qw8d gl*5lbjn.1e-hythe number of new model iPhone's sold were recorded for thel dhb1y.jnemw8 i/;s qlg-5 *x first week after the phone's release. The following table shows this data.

1. Draw a scatter diagram for this data on the following axes. Use a scale of 1 unit to represent each day on the x -axis and 1 unit to represent smartphones on the y -axis.


2. Use your graphic display calculator to find
1. the mean number of days, $\bar{x}$ ;   
2. the mean number of new model iPhone's, $\bar{y}$ .≈   
3. Plot the point $M(\bar{x}, \bar{y})$ on your scatter diagram.
4. Calculate the Pearson's product-moment correlation coefficient, r .≈   
5. Find the equation of the regression line y on x for this week.y =ax+b a =    b =   
6. Draw the regression line from part (e) on your scatter diagram.
7. Use the line of regression equation to estimate the number of new model iPhone's sold on the 8th day after release.≈   
8. State whether your estimate from part $\mathbf{( g )}$ is reliable. Justify your answer.

  The following table shows the number of chin-ups and the number of phl f0u8c9cgo7 ush-ups perforu8g ocl709f chmed by seven college athletes.

Draw a scatter diagram for the above data on the axes below. Use a scale of 1 unit to represent 2 chin-ups on the x-axis and 1 unit to represent 10 push-ups on the y-axis.





2. 1. Find the Pearson's product-moment correlation coefficient, r , for the above data. ≈   
2. Describe the correlation between the number of chin-ups and the number of push-ups.
3. Find the
1. mean number of chin-ups, $\bar{x}$ ;≈   
2. mean number of push-ups, $\bar{y}$ .≈   
4. Plot and label the point $\mathrm{M}(\bar{x}, \bar{y}) $ on your scatter diagram from part $(a)$.
5. Find the line of regression equation y on x .y =ax+b a =    b =   
6. Use your equation from part (e) to estimate the number of push-ups of a college athlete who performed 14 chin-ups.≈   
7. Draw the regression line y on x on your scatter diagram from part $(a)$.

A new college athlete performs 30 chin-ups and uses the above data to estimate that he will perform 96 push-ups.
8. State whether this estimate is reliable and give a reason for your answer.

  A group of eight undergraduate engineering studentsn z9:pa,sn(kv2(+tmqeo 8ed e were surveyed to determine their satisfaction with their semester 1 mathematics course. The following table 9vtps:(qe,o82(ednne maz+ kshows their responses using a seven-point scale, and their final grades for this course.


1.On the axes below, draw a scatter diagram for this data. Use a scale of 1 unit to represent ratings on the x-axis and 1 unit to represent 5 grade percent on the y-axis.


2. 1. Calculate the Pearson's product-moment correlation coefficient for this data, r .≈   
2. Describe the correlation between students' mathematics course grades and survey results.
3. Calculate
1. the mean rating, $\bar{x}$ ;≈   
2. the mean grade, $\bar{y}$ .≈   
4. Plot the point $\mathrm{M}(\bar{x}, \bar{y})$ on your scatter diagram.
5. Find the line of regression equation, y on x .y =ax+b a =    b =   
6. Draw the regression line from part $(e)$ on your scatter diagram.
7. Use the line of the regression to estimate the final grade for the student who gave a 7 rating for the course.≈    %
8. State whether your estimate is reliable and justify your answer.

  The amount of dry matter in tomatoes, measured in grams, is the amount of sol-2g hmwm l; su3z2j ,rx;ufi;pid content of the fruit, subtract the water content. The table below sh;;m pfs i l22z;hg j-,uumxw3rows weights of dry matter in 10 tomatoes grown in soil with different amounts of calcium, Ca, measured in $\mathrm{mg} / \mathrm{cm}^{3}$ .



1. 1. Find the Pearson's product moment correlation coefficient, r , for this data.≈   
2. Comment on the strength of the r value found.
2. 1. Find the Spearman's rank correlation coefficient, $r_{s}$ , for this data.≈   
2. Comment on the strength of the $r_{s}$ value found.
3. In the context of this question and the data provided, comment on if both correlation coefficients should be considered when determining the relationships between soil calcium content and dry matter weight, or if one should be used in favour of the other.

  The table below shows the lengths of at01 2aw xflv0omhlete's legs, x (in metres), and the time, y (in seconds), these athletes recorded in the 100 metre sprint during the high school state champiol 2fox0wva1 m0nships.



1. Find the range of the sprint times for these athletes.=    seconds
2. For the given data
1. calculate the Pearson's product-moment correlation coefficient, r ; ≈   
2. describe the correlation between the leg length and sprint times of these athletes.
3. Use your graphic display calculator to find the line of regression equation y on x , in the form y=m x+c

Another athlete, Henry, has a leg length of 0.82 $\mathrm{~m}$ ..y =ax+b a =    b =   
4. Use your line of regression equation to estimate the time Henry records for his 100 $\mathrm{~m} $ sprint, correct to two decimal places.

Henry actually recorded a time of 16.5 seconds.≈    seconds
5. Find the percentage error in your estimate in part (d).

Travis, a shorter athlete, has a leg length of 0.76 $\mathrm{~m}$ .≈    %
6. State whether it is valid to use your regression line in part (c) to estimate the time Travis can sprint the 100 $\mathrm{~m}$ . Give a reason for your answer.
7. If the Spearman's rank correlation coefficient were to be calculated on the original 8 athletes, state Oliver's x and y rank.

  Particulate Matter (PM) is a measure of air pollution. PM2.5 ,z;2fan,l jn iconcentration, measured,na, n;fjli2z in $\mu \mathrm{g} / \mathrm{m}^{3}$ , is a measure of the amount of fine, inhalable particles with diameters of 2.5 micrometres or smaller in the air. The PM2.5 levels were measured at various times on a particular day in Delhi, India, with the results shown in the table below. The max and min PM2.5 levels were recorded and are are included in the data.



1. Determine an appropriate model for the PM2.5 concentration, P , in terms of the number of hours after midnight, t .

On this particular day, the Central Pollution Control Board of Dehli alerts citizens that the air quality will be "very poor" between 09:15 to 10:15 and recommends that the citizens should minimise being outside during these times of the day.$P(t)=a \sin (b-c)+d$ a =    b =    c =   d =   
2. Use your model from part (a) to estimate the PM2.5 level that the Central Pollution Control Board considers to be "very poor".$\approx$   $\mu \mathrm{g} / \mathrm{m}^{3}$

  The London National Theatre is presenting the famous Shakespearean play Hamlet. oic5uc9r9( woz2s(d jia:qus. eg * *oMatthew is interested in the age range of the audience attendees and collects data by standing at the entrance and sampling every fifth audience member as they enter the theatre. Using this data, he constructs a box-and-whisker diagram to represent the data, as shown uc c*iuoo 2dj*s59qw9ire( :goz.(a sbelow.


1. State the sampling method Matthew used in his study.
2. Write down the median age of the sample.
3. Calculate the interquartile range.

The youngest audience member sampled, Rosie, is just 15 years old. Matthew believes that Rosie is an outlier and is considering excluding her from the sample data set.    years
4. Determine if Matthew is correct. Give a reason to your answer.

When asking their ages, Matthew also asks how many other Shakespearean plays they have watched in their lifetime ( y ). The following scatter diagram shows this data, paired with their age (x) .



5. Describe the correlation between the two variables.

Matthew used the data to calculate the equation of the regression line y on x as

y=0.458 x-5.76


Matthew used this equation to estimate the number of Shakespearean plays a 65 year old audience attendee would have previously watched.
6. 1. Determine Matthew's estimate.≈   
2. State whether it is valid to use the line of regression equation for this estimate. Give a reason for your answer.

Matthew, an avid Shakespearean fan, wants to understand if there is a relationship between his favourite Shakespearean plays and their length, measured by total run time in minutes. He creates a table, shown below, and ranks 8 Shakespearean plays (A-H), their run time length, and his rating out of all the Shakespearean plays he has seen ( 1 being his favourite).



For this analysis, Matthew uses the Spearman's rank correlation coefficient, r s to understand the relationship between the variables. A rank table is created.




7. Copy and complete the rank table.
8. 1. Calculate the value of $r_{s}$ .   
2. Interpret the result.

  The following table show 6-ct/,:aq8siscmpj(lnqoa/ s the total revenue, y, in Australian dollars (AUD),
obtained daily during the first week of January 2020, by Peppy's Pizza restaurant and the number of guests, x, served.



1. 1. Calculate the Pearson's product-moment correlation coefficient, r , for this data.    -
2. Hence comment on the result.
2. Write down the equation of the regression line y on x .y = ax+b a =    b =   
3. Use the line of the regression to estimate the revenue of serving 70 guests. Give your answer correct to the nearest AUD.

Daily maintenance cost for the restaurant is 240 AUD. Additionally, the cost of serving one guest is 5 AUD.≈    AUD
4. Determine if the restaurant makes a profit when serving 45 guests on a particular day.
5. 1. Write down an expression for the total revenue of serving x guests.ax+b a =    b =   
2. Find an expression for the profit of the restaurant when serving x guests on a particular day.ax+b a =    b =   
3. Find the least number of guests required to be served to result in a profit for the day.

  The following table shows the total revenue, y, in US dollars (USD), obtained me)0tc:dj f ,o x*ha1jj9dqm; ponthly duaj1,j;et*jx d09f)dm: pocqhring the first six months of 2020, by Law Office of Fox Brothers and the number of clients, x, served.




1. 1. Calculate the Pearson's product-moment correlation coefficient, r , for this data.   
2. Hence comment on the result.
2. Write down the equation of the regression line y on x .y =ax+b a =    b =   
3. Use the line of the regression to estimate the revenue of serving 20 clients. Give your answer correct to the nearest USD.

Monthly operating cost for the law office is 2500 dollars. Additionally, the cost of serving each client is 200 dollars.
4. Determine if the law office makes a profit when serving 6 clients on a particular month.   
5. 1. Write down an expression for the total revenue of serving x clients. ax+b a =    b =   
2. Find an expression for the profit of the law office when serving x clients on a particular month. ax+b a =    b =   
3. Find the least number of clients required to be served to result in a profit for the month.

  The following table shows the average height, x, and the av 9zbqe 0hcd42 mqp4fl m;gb9z3erage trunk diameter, y, of eight species of tree mbdlg099;qb z34ef4 hpqcmz2.


1. Find the range of the average heights for these eight species of tree, correct to two decimal places.   
2. From the data from of these eight species of tree
1. calculate the Pearson's product-moment correlation coefficient, r ; ≈   
2. describe the correlation between the average height and the average trunk diameter.
3. Write down the line of regression equation, y on x , in the form y=m x+c .

The average height of another tree species, Black Ash, is 15.2 $\mathrm{~m}$ .y =ax+b a =    b =   
4. Use your regression line to estimate the average trunk diameter of Black Ash.

The average diameter of Black Ash is found to be 0.445 $\mathrm{~m} $. ≈   
5. Find the percentage error in your estimate in part (d).

The average height of another tree species, Alpine Ash, is 87.5 $\mathrm{~m}$ . ≈    %
6. State whether it is valid to use the regression line to estimate the average trunk diameter of Alpine Ash. Give a reason for your answer.
7. Complete the two empty rank columns in the table and state why the Spearman's rank correlation coefficient could be calculated by inspection.

  The tables below show the grip strength of volunteer, measured in kilogram- sf t7xu*8d12wo q:sz-y1 lhxtforce (kgf dxfzwots-8yu2qxs 11 l t:*7h) with a dynamometer, at various times of a certain day starting at midnight.


1.Draw the scatter plot on the axes provided below and describe it.


2. Find an appropriate model for the volunteers grip strength S in terms of hours since midnight h .
3. Using your model, find the average grip strength of the volunteer.
4. Use your model to predict the volunteers qrip strenath at 15: 00 .≈    kgf

  The Elite Traveller and Condé Nast Tr u0.xyqvzb1 5aaveller are two magazines that review and rank the best restaurants in Paris. Based on different sets of criteria, scores out of 10 are assigned to 8 restaurants in Paris 1yvabuz5xq. 0(labelled A to H). The raw data is collated in the following table.


1. 1. Find the Pearson's product-moment correlation coefficient, r , for this data.≈   
2. Using the value of r , interpret the relationship between the Elite Traveller's score and the Conde Nast Traveller' score.
2. Find the equation of the line of regression, y on x .y =ax+b a =    b =   
3. 1. Use your regression equation from part (b) to estimate the Condé Nast Traveller's score when the Elite Traveller awarded a perfect 10.≈   
2. State whether this estimate is reliable. Justify your answer.
4. Complete the two empty rank rows in the table above.
5. 1. Find the value of the Spearman's rank correlation coeffcient, $r_{s} $.≈   
2. Comment on the result obtained for $ r_{s}$ .

The Director of the Elite Traveller Magazine believes that the score for restaurant E was too high and decreased the score from 9.7 to 9.2 .
6. Explain why the value of the Spearman's rank correlation coefficient $r_{s} $ does not change.

  A High School in Auckland, New Zealand, conducted a study on58c hlkcm 7 6w1 s6kg6q.xdw5gbv8mor the relationship between teaching experience, in years, and teaching performance. There were 8 teachers involved in the study (labelled A to H). Teaching performmm.s5c 6lkvqwrd xcb8g og58k1 67w6hances were assessed by the commission board of the school on a scale out of 100. The data is shown in the following table.


1. 1. Find the Pearson's product-moment correlation coefficient, r .≈   
2. Using the value of r , interpret the relationship between teaching performance and teaching experience of these teachers.
2. Write down the equation of the regression line y on x .y =ax+b a =    b =   
3. 1. Use your regression equation from part (b) to estimate the teaching performance score of a teacher who has 8 years of teaching at the school.≈   
2. State whether this estimate is reliable. Justify your answer.
4. Complete the two empty rank rows in the table above.
5. 1. Find the value of the Spearman's rank correlation coeffcient, r_{s} .≈   
2. Interpret the result obtained for r_{s} .

This data collected excluded the Teachers first year of teaching, as they were on probation and not fully registered yet. The commission board decided to include this period to the working experience.
6. Explain why the value of the Spearman's rank correlation coefficient r_{s} does not change when this first year is included.

  Kristina recently op2vy dom7um9 v3:bg )vtened a painting exhibition in Berlin. She wants to estimate the average number of guests coming to the gallery per day after the opening. The tablevy 2u 3g:tv mdbv97mo) below shows some data of days that Kristina did record how many guests came to the gallery.



1. Find a linear model of $\ln (G)$ in terms of d .$\ln (G)=a(d)+b$ a =    b =   
2. Hence or otherwise, find an exponential model of G in terms of d .
3. Using one of your models, estimate how many guests came to the gallery on the opening day and explain why this estimate could be inaccurate.

A household appliance manufacturing company uses th2esy35:gv) x5lev ofpe following model to predict the quantity of output, Q (in thousands of p)3v5gysefe lox5v 2:items), from the quantity of labour, l (in hundreds of hours).

$Q(l)=a l e^{b l}, a, b \in \mathbb{R}$

The production analysts in the company are trying to decide between two options for the values of a and b , depending on which historical time period to use as their data set. The two options, Models A and B are as follows:
Model A: a=4, b=-0.3
Model B: a=3.5, b=-0.2

To test whether model A or B has better capacity of forecasting output, the company records the quantity of output for three different quantities of labour, as shown in the table below.



Determine the model that the firm should choose based on the least sum of square of residuals.

Phillip, an IB math teacher, h6jy0(htb a6db as formulated a mathematical model to predict the average mark on the final exam of the students, M , from the time allowj6dba0ty6b h( ed to complete the exam, h , in hours. The model is described as follows.

$M(h)=c h e^{d h}, c, d \in \mathbb{R}$

Phillip is trying to determine the values for c and d and two options, depending on which prior class data to use. His options are Model A or Model B , as follows:

$\begin{array}{l}
\text { Model A: } c=102, d=-0.48 \\
\text { Model B: } c=110, d=-0.55
\end{array}$

To determine whether Models A or B is more accurate in predicting the average exam mark, Philip conducts three trial examinations with his class with different exam lengths. The results are shown in the table below.


Phillip decides to choose the model with the smallest value of the sum of square of residuals. Determine the model that he should choose.

  During a trip to a forest to forage for mushroomrd.-r e0vily (s, Viviane finds a giant mushroom. She decides to model the shdle0vr(y.r -i ape of the mushroom to find its volume.

After taking a photo of the mushroom and zooming in to get its real size, she rotates the photograph and estimates that the cross-section passes through the points (0,3),(15,3) , (15,15),(22,10),(23.5,6) and (24,0) , where all measurements are in centimetres. The cross-section is symmetrical about the x -axis, as shown below.


Viviane models the section from (0,3) and (15,3) with a straight line.
1. Write down the equation of the line that passes through these points.

Next, Viviane models the section that passes through the points (15,15),(22,10),(23.5,6) and (24,0) with a quadratic curve.   
2. 1. Use your G.D.C. to find the equation of this quadratic curve.y=$ax^b+cx+d$ a =    b =    c =    d =   
2. By considering the gradient of the curve at the point (15,15) , explain why this may not be a good model.

Viviane thinks she can obtain a better model if a quadratic passing through the point (24,0) with a maximum point at (15,15) is used.
3. Find the equation of this model, in the form $y=a(x-h)^{2}+k $.

Using this new model, Viviane proceeds to estimate the volume of the mushroom by finding the volume of revolution about the x -axis.y ≈ a(x+b)^c+d a =    b =    c =    d =   
4. 1. Write down an expression for her estimate as a sum of two integrals.
2. Find the volume of the mushroom estimated by Viviane.≈    $cm^3$

  Lohan receives an antique vase as a gifsdkb6k-:5 s-skul+ oo. c)y5zuhjrb4t from her grandparents. She decides to model the shape of the vase to calculate itkuzs.)c- 4:k djkh6-r bs5 o5uoy b+lss volume.

She places the vase horizontally on a piece of paper and uses a pencil to mark five representative points (0,5),(7.5,10),(17,7.5),(25,3) and (35,7.5) , as shown below. These points are connected to form a symmetrical cross-section about the x -axis. All units are in centimetres.


Lohan initially uses a straight line to model the section from (25,3) to (35,7.5) .
1. Determine the equation of the line that passes through these two points.

Lohan thinks that a quadratic curve might be a good model for the section from (0,5) to (25,3) . She carries out a least squares regression using this model for the points she has recorded.y =ax+b a =    b =   
2. 1. Determine the equation of the least squares regression quadratic curve found by Lohan.y≈ $ax^b+cx+d$ a =    b =    c =    d =   
2. By considering the gradient of the curve at the point where x=10 , determine whether the quadratic regression curve is a good model or not.

Lohan decides that a cubic curve for the entire section from (0,5) to (35,7.5) would be a better fit.
3. Find the equation of the cubic model.

Using this model, Lohan estimates the volume of the vase by calculating the volume of revolution about the x -axis.y≈ $ax^b+cx^d+ex+f$ a =    b =    c =    d =    e =    f =   
4. Find the volume of the vase estimated by Lohan.

Lohan subsequently fills the vase with water and discovers that the true volume is 5500 $cm^3$ . ≈    $cm^3$
5. Calculate the percentage error in Lohan's estimate of the volume.    %

  Every year a sports academy enrols new members faehwljgdk/ w 3;r9l:5rom a large group of applicants. This year the academy enrols 1009 e5kwjdg3w/rah ;l: l 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.

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.≈   
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.

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.

  The table below shows the daily revenue, R, in thousands of USD, of a toyn;jspf42 c4* 1xn jtey store during the month of December;*netxj4s2 cyf4 np1 j. The first row of the table shows the day of the month (t) and the second row the revenue of that day in thousands of USD.


1.1.Generate a scatter diagram for this data on the axes below.



2. Identify a possible outlier and give a possible reason for this outlier;
3. Give a reason why plotting R vs $\log (t)$ is likely to be the best linearisation method.
2. Linearise the data and determine an appropriate linear model for the daily revenue R in terms of day of the month in December, t .
3. Using your model, predict the revenue after 3 weeks in December. ≈    thousands of USD
4. The total revenue for the month of December can be estimated by $\int_{a}^{b} R \mathrm{dt}$ .
1. Write down the values of a and b . a =    b =   
2. Estimate the total revenue in December using your model.≈    thousands of USD

The following table lists the gold medal winners in the male javelin throwi 6up-t,7yjy5xc2 gt7 ybqj d)nng competition in a youth athletics event. The table shows the winner in each age category and their associated longest jjdtubyj5 x2y7-)t,nc 7y g6pqavelin throw.

1.Sketch a scatter diagram for the data on the axes below and describe it.


2. Determine an appropriate non-linear model using linearisation with logarithms to model distance thrown, d , in terms of age, a .
3. Find the coefficient of determination for your model and interpret.
4. Using your model, determine the athlete(s) who, given their age, threw the javelin
1. longer than expected.
2. shorter than expected.

  The table below shows the average number of words of a child's vocabulary a*w/f 3 mhf vymsdmz(g/, yr26ls age increases on a monthly basis. The first row shows the child's age in months (tv mzh/mfg /2dw,ym3ly s* fr6(), and the second row shows the average number of words, W, in hundreds.


1.Generate a scatter diagram for this data on the axes below and describe the trend.


2. Linearise the data and determine an appropriate linear model for the number of words, W , in terms of the months, t .$W=a \log (\mathrm{t})+b$ a =    b =   
3. Using your model, predict the approximate number of words in the vocabulary of a child who has just turned 6 years old.

The rate of change for the number of words a child learns per month can be estimated by W^{\prime}(t) .
4. Estimate the month at which the child learns 100 words per month.≈    month

  John operates a small clothing factory that man3a8bn3(l/t4 hap0zrqo omx eyz1 w.x; ufactures jeans. John observes that the weekly total production cost, C , in Australian dollars (AUD), and the number of jeans produced per week, N , can be related by the equation 3em/y .bxozrx8 ph o41w3t0qa(nlz;a

$C=a N^{b}+K$,

where a, b and K are positive constants.
John estimates that the weekly total fixed cost of operating the factory is 7500 AUD.
1. Write down the value of K .

After analysing the financial accounting records of a particular month, John finds the data given below.   

2. Draw a scatter diagram of $\ln (C-K)$ versus $\ln N$ , scaling and shifting the axes if needed.
3. State the type of model that best fits the data displayed on your scatter diagram from part (b).
4. Write down the equation of the regression line of $\ln (C-K) $ on $\ln N$ .
5. Hence find the value of a and the value of b .

John wants to increase the production rate of jeans up to 1000 pairs per week. a =    b =   
6. Using John's equation, estimate the weekly total cost of producing 1000 jeans. ≈    AUD
7. State whether it is valid to use John's equation to estimate the weekly total cost of producing 1000 jeans. Give a reason for your answer.
8. 1. Describe how the data must be entered into your G.D.C. to determine John's equation using power regression method.
2. Hence verify your answers to part (e). a =    b =   

  Sandra operates a pizzeria that has 24 hours delivery service. Sandra thi*7*ew t 9mrybpnks that the dailbr 9ey*m *ptw7y total production cost, C , in euros (EUR), and the number of pizzas made per day, N , can be related by the equation

$C=a N^{b}+K$,

where a, b and K are positive constants.
Sandra estimates that the daily total fixed cost of operating the pizzeria is 1000 EUR.
1. Write down the value of K .

After analysing the bookkeeping records of a particular working week, Sandra finds the data given below.   

2. Draw a scatter diagram of $\log _{2}(C-K)$ versus $\log _{2} N$ , scaling and shifting the axes if needed.
3. State the type of model that best fits the data displayed on your scatter diagram from part (b).
4. Write down the equation of the regression line of $\log _{2}(C-K)$ on $\log _{2} N$ .
5. Hence find the value of a and the value of b .

Sandra wants to increase the selling rate of pizzas up to 800 items per day. a =    b =   
6. Using Sandra's equation, estimate the daily total cost of producing 800 pizzas.≈   
7. State whether it is valid to use Sandra's equation to estimate the daily total cost of producing 800 pizzas. Give a reason for your answer.
8. 1. Describe how the data must be entered into your G.D.C. to determine Sandra's equation using power regression method.
2. Hence verify vour answers to nart (e) a =    b =