HLAA Bivariate Analysis [34 marks]
1.
[Maximum mark: 5]
21M.1.SL.TZ1.4
A research student weighed lizard eggs in grams and recorded the results. The
following box and whisker diagram shows a summary of the results where L
and U are the lower and upper quartiles respectively.
The interquartile range is 20 grams and there are no outliers in the results.
(a)
Find the minimum possible value of U .
[3]
(b)
Hence, find the minimum possible value of L.
[2]
2.
[Maximum mark: 15]
EXM.2.SL.TZ0.1
The principal of a high school is concerned about the effect social media use
might be having on the self-esteem of her students. She decides to survey a
random sample of 9 students to gather some data. She wants the number of
students in each grade in the sample to be, as far as possible, in the same
proportion as the number of students in each grade in the school.
(a)
State the name for this type of sampling technique.
[1]
The number of students in each grade in the school is shown in table.
(b.i)
Show that 3 students will be selected from grade 12.
[3]
(b.ii)
Calculate the number of students in each grade in the sample.
[2]
In order to select the 3 students from grade 12, the principal lists their names in
alphabetical order and selects the 28th, 56th and 84th student on the list.
(c)
State the name for this type of sampling technique.
Once the principal has obtained the names of the 9 students in the random
sample, she surveys each student to find out how long they used social media
the previous day and measures their self-esteem using the Rosenberg scale. The
Rosenberg scale is a number between 10 and 40, where a high number
represents high self-esteem.
[1]
(d.i)
Calculate Pearson’s product moment correlation coefficient, r.
[2]
(d.ii)
Interpret the meaning of the value of r in the context of the
principal’s concerns.
[1]
(d.iii) Explain why the value of r makes it appropriate to find the
equation of a regression line.
[1]
(e)
Another student at the school, Jasmine, has a self-esteem value
of 29.
By finding the equation of an appropriate regression line,
estimate the time Jasmine spent on social media the previous
day.
[4]
3.
[Maximum mark: 7]
23M.2.SL.TZ1.4
The total number of children, y, visiting a park depends on the highest
temperature, T , in degrees Celsius (°C). A park official predicts the total number
of children visiting his park on any given day using the model
y = −0. 6T
(a)
2
+ 23T + 110, where 10 ≤ T ≤ 35.
Use this model to estimate the number of children in the park
on a day when the highest temperature is 25 °C.
[2]
An ice cream vendor investigates the relationship between the total number of
children visiting the park and the number of ice creams sold, x. The following
table shows the data collected on five different days.
Total number
of children (y)
Ice creams
sold (x)
(b)
(c)
81
175
202
346
360
15
27
23
35
46
Find an appropriate regression equation that will allow the
vendor to predict the number of ice creams sold on a day when
there are y children in the park.
[3]
Hence, use your regression equation to predict the number of
ice creams that the vendor sells on a day when the highest
temperature is 25°C.
[2]
4.
[Maximum mark: 7]
SPM.2.AHL.TZ0.4
The following table below shows the marks scored by seven students on two
different mathematics tests.
Let L1 be the regression line of x on y. The equation of the line L1 can be written in
the form x = ay + b.
(a)
Find the value of a and the value of b.
[2]
Let L2 be the regression line of y on x. The lines L1 and L2 pass through the same
point with coordinates (p , q).
(b)
Find the value of p and the value of q.
[3]
(c)
Jennifer was absent for the first test but scored 29 marks on the
second test. Use an appropriate regression equation to estimate
Jennifer’s mark on the first test.
[2]
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