6.1 Extended Questions (Paper 2, SL)

A study was conducted on  participants, measuring their body fat percentage (%) and their resting heart rate in beats per minute).  The results are shown in the table below.

Body fat percentage
Resting heart rate

Use your graphic display calculator to find

, the mean body fat percentage

, the mean resting heart rate

, the Pearson’s product-moment correlation coefficient.

Write down the equation of the regression line of  on for this data, giving your answer in the form    where  and   are constants to be found.

Show that the point lies on the regression line of   on .

A seventh participant, John, has a resting heart rate of .

Use the regression line equation to estimate John’s body fat percentage.

Justify whether it is valid to use the regression line of  on  to estimate John’s body fat percentage.

John’s body fat percentage is .

Calculate the percentage error in John’s estimated body fat percentage from part (c).

A farm is shown in the diagram below.  A motorway runs in a straight line along the edge of the farm from point  to point , and the farmhouse is located at point .    and  form the other two sides of the farm, and the distances from the farmhouse to points  and  are  and  respectively.  Angle  is , and points ,  and  lie in a horizontal plane.

Calculate the distance along the motorway from  to .

The cost of fencing in US dollars ) is $  per metre.

Calculate the total cost of fencing the whole perimeter of the farm. Give your answer to decimal places.

Calculate the area of the farm.

Find the sizes of angles and .

Calculate the shortest distance from the farmhouse to the motorway.

A vertical signpost is located at point , and the top of the signpost is designated as point .  The angle of elevation to the top of the signpost from point  is measured to be .

Calculate the distance , the vertical height of the signpost.

Calculate the distance between the top of the signpost, , and point .

The rate of growth of the grass on the farm, , in inches per month, can be modelled by the function

where  is the temperature in degrees Fahrenheit.

Find the maximum rate of grass growth on the farm and the temperature required.

The table below shows the distribution of the number of baskets scored by netball players during a weekly game.

Number of baskets
Frequency

Calculate

the mean number of baskets scored by a player

Find the median number of baskets scored.

Find the interquartile range.

Determine if a player who scored baskets would be considered an outlier.

Two players are randomly chosen.

Given that the first player scored  or less baskets, find the probability that both players scored exactly  basket

The number of hours each player trains each week is normally distributed with a mean of  hours and standard deviation of  hours.

Calculate the probability that a player trains less than  hours a week.

Calculate the probability that a player trains less than hours a week.

Calculate the expected number of players that train between and  hours a week.

Best Beans is a New Zealand-based company that sells baked beans packaged in cylindrical cans.

Given that their cans have a height of  and a diameter  of calculate

(i)      the volume of the can

Every month, Best Beans expects to sell  thousand cans of baked beans.  It is known that

where  is the monthly profit, in New Zealand dollars , from the sale of  thousand cans of baked beans.  It is also known that Best Beans makes a profit of in a month where it sells  cans of baked beans.

Find .

Find the least number of cans which must be sold each month in order to make a profit.

Find the monthly sales level that will maximise profit, and the expected profit at this level.

Best Beans wants to buy a new factory at a cost of .  The CEO decides to invest  of the company’s monthly profit into a savings account paying a nominal annual interest rate of  compounded monthly.

Under the assumption that the company’s monthly profit will attain its maximum value every month throughout the period, determine whether Best Beans will have saved enough to buy the factory by the end of two years.

 people are asked if they like juice , tea  and/or coffee  for breakfast.

 like all three

 like coffee and tea

 like coffee and juice

 like juice only

 like coffee only

 like tea only

(i)     Draw a Venn diagram to represent the information provided.

(ii)    Write down the number of people who like coffee but not tea.

There are  people in total who like tea.

(i)     Calculate the number of people who like tea or juice.

(ii)    Find the number of people who like none of the drinks for breakfast.

A person is chosen at random from the  people.

Find the probability that this person

(i)      likes coffee

 likes coffee and tea but not juice

 does not like either tea or juice