Problem Bank - Bivariate Statistics

Access custom-built, exam-style problems for bivariate statistics. Each problem has a full solution and mark-scheme, as well as AI grading and support.

Select a Difficulty:

IB: 4

IB: 5

IB: 6

0 / 8

Dr. Alvarez, a nutrition researcher, believes there is an association between a patient’s daily sugar intake, X (in grams), and their fasting blood‑glucose level, Y (in mg/dL). He tests his family and friends and the following paired data points are collected.

Patient	X (g)	Y (mg/dL)
1	50	95
2	60	102
3	70	98
4	80	110
5	90	105
6	100	100
7	110	108
8	120	97
9	130	101
10	140	109
11	150	95
12	160	106

Write down the sampling method used by Dr. Alvarez.
[1]
State one drawback of this sampling method.
[1]
State suitable hypotheses H0 and H1 for a two-tailed test of Dr. Alvarez's claim.
[1]
Carry out the test at the 5% significance level. With reference to the p-value, state your conclusion in the context of Dr. Alvarez's claim.
[4]

Dr. Alvarez fits the regression line of Y on X as Y^=0.50X+80.0.

She uses this to predict that her brother, who consumes 150g of sugar will have a fasting glucose of 155mg/dL.

Comment on the validity of this prediction with a mathematical justificiation.
[1]

0 / 9

A reaction is studied in a school chemistry laboratory. The production rate R (kg/h) of a chemical is modeled as a cubic in the reactant concentration c (mol/L):

R(c)=kc3+mc2+nc+p

For four test runs, the technician recorded the concentration and the time to produce 1kg of the chemical:

Concentration c (mol/L)	0.50	1.00	1.50	2.00
Time to produce 1 kg (h/kg)	1.701	0.833	0.415	0.217

Using the table, one can compute the rate R (kg/h) at each concentration by inverting the “time per kg”.

Use a cubic regression to determine k,m,n, and p and state the model.
[3]
Using your model, estimate the time (in minutes) to produce 1kg when c=1.80 mol/L.
[3]
For what values of c in the range 0.50≤c≤2.00 does the model predict a rate of at least 3.0 kg/h?
[3]

0 / 14

A species of invasive fish is introduced into a lake. The local environmental agency records the following total number of fish, N, over the first few days (t in days):

t	1	2	3	4	5	6
N	25	42	73	126	218	377

An exponential model of the form N=abt is proposed.

Use exponential regression to find the values of a and b, correct to 4 decimal places.
[3]
Hence write down the coefficient of determination.
[1]
Using this model, estimate the number of new fish that appear on day 7.
[2]

In fact, the population does not grow indefinitely. Biologists instead propose a logistic model of the form

N=1+ce−ktL,

with L=1200.

Using the data from day 6 (N=377) and day 12 (1010), find the values of c and k, correct to 3 significant figures.
[5]
Using your logistic model, determine the day when the population is increasing at the fastest rate.
[3]

0 / 14

The following table displays values of lnx and lny.

lnx	0	2	5.1	7	8
lny	4	3	1.5	0.6	0.1

Find the value of x when y=e4.
[1]

The relationship between lnx and lny can be modelled by the regression equation lny=alnx+b.

Using a graphic display calculator, find the value of a and the value of b.
[3]
Hence estimate the value of x when y=2.7.
[4]
Explain why this model should not be used to predict the value of lny when x=0.1.
[2]

The relationship between x and y can be modeled by the equation y=kxn.

Find the value of k and the value of n.
[4]

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Select a Difficulty:

IB: 4

IB: 5

IB: 6

0 / 8

Patient	X (g)	Y (mg/dL)
1	50	95
2	60	102
3	70	98
4	80	110
5	90	105
6	100	100
7	110	108
8	120	97
9	130	101
10	140	109
11	150	95
12	160	106

Write down the sampling method used by Dr. Alvarez.
[1]
State one drawback of this sampling method.
[1]
State suitable hypotheses H0 and H1 for a two-tailed test of Dr. Alvarez's claim.
[1]
Carry out the test at the 5% significance level. With reference to the p-value, state your conclusion in the context of Dr. Alvarez's claim.
[4]

Dr. Alvarez fits the regression line of Y on X as Y^=0.50X+80.0.

She uses this to predict that her brother, who consumes 150g of sugar will have a fasting glucose of 155mg/dL.

Comment on the validity of this prediction with a mathematical justificiation.
[1]

0 / 9

A reaction is studied in a school chemistry laboratory. The production rate R (kg/h) of a chemical is modeled as a cubic in the reactant concentration c (mol/L):

R(c)=kc3+mc2+nc+p

For four test runs, the technician recorded the concentration and the time to produce 1kg of the chemical:

Concentration c (mol/L)	0.50	1.00	1.50	2.00
Time to produce 1 kg (h/kg)	1.701	0.833	0.415	0.217

Using the table, one can compute the rate R (kg/h) at each concentration by inverting the “time per kg”.

Use a cubic regression to determine k,m,n, and p and state the model.
[3]
Using your model, estimate the time (in minutes) to produce 1kg when c=1.80 mol/L.
[3]
For what values of c in the range 0.50≤c≤2.00 does the model predict a rate of at least 3.0 kg/h?
[3]

0 / 14

A species of invasive fish is introduced into a lake. The local environmental agency records the following total number of fish, N, over the first few days (t in days):

t	1	2	3	4	5	6
N	25	42	73	126	218	377

An exponential model of the form N=abt is proposed.

Use exponential regression to find the values of a and b, correct to 4 decimal places.
[3]
Hence write down the coefficient of determination.
[1]
Using this model, estimate the number of new fish that appear on day 7.
[2]

In fact, the population does not grow indefinitely. Biologists instead propose a logistic model of the form

N=1+ce−ktL,

with L=1200.

Using the data from day 6 (N=377) and day 12 (1010), find the values of c and k, correct to 3 significant figures.
[5]
Using your logistic model, determine the day when the population is increasing at the fastest rate.
[3]

0 / 14

The following table displays values of lnx and lny.

lnx	0	2	5.1	7	8
lny	4	3	1.5	0.6	0.1

Find the value of x when y=e4.
[1]

The relationship between lnx and lny can be modelled by the regression equation lny=alnx+b.

Using a graphic display calculator, find the value of a and the value of b.
[3]
Hence estimate the value of x when y=2.7.
[4]
Explain why this model should not be used to predict the value of lny when x=0.1.
[2]

The relationship between x and y can be modeled by the equation y=kxn.

Find the value of k and the value of n.
[4]

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Patient	X (g)	Y (mg/dL)
1	50	95
2	60	102
3	70	98
4	80	110
5	90	105
6	100	100
7	110	108
8	120	97
9	130	101
10	140	109
11	150	95
12	160	106

Patient	X (g)	Y (mg/dL)
1	50	95
2	60	102
3	70	98
4	80	110
5	90	105
6	100	100
7	110	108
8	120	97
9	130	101
10	140	109
11	150	95
12	160	106

Patient	X (g)	Y (mg/dL)
1	50	95
2	60	102
3	70	98
4	80	110
5	90	105
6	100	100
7	110	108
8	120	97
9	130	101
10	140	109
11	150	95
12	160	106

Patient	X (g)	Y (mg/dL)
1	50	95
2	60	102
3	70	98
4	80	110
5	90	105
6	100	100
7	110	108
8	120	97
9	130	101
10	140	109
11	150	95
12	160	106

Patient	X (g)	Y (mg/dL)
1	50	95
2	60	102
3	70	98
4	80	110
5	90	105
6	100	100
7	110	108
8	120	97
9	130	101
10	140	109
11	150	95
12	160	106

Patient	X (g)	Y (mg/dL)
1	50	95
2	60	102
3	70	98
4	80	110
5	90	105
6	100	100
7	110	108
8	120	97
9	130	101
10	140	109
11	150	95
12	160	106