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Inferential Statistics · Paper 1 — Free Maths AI HL Practice Questions

1FoundationSAQ-SRegression analysis5 marksPaper 1~8 min

Data

b = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2}, \qquad a = \bar{y} - b\bar{x}

The heights (in cm) and weights (in kg) of six adult males are recorded below. Height (

x\,\text{cm}

) — 165 — 170 — 175 — 180 — 185 — 190 Weight (

y\,\text{kg}

) — 62 — 68 — 72 — 78 — 82 — 88

(a)

State the type of regression that would be most appropriate for modelling the relationship between height and weight. [1 mark]

(b)

Find the equation of the regression line of weight on height in the form

y = a + bx

, giving the values of

a

and

b

correct to three significant figures. [4 marks]

Solutions

2MasterySAQ-SRegression analysis5 marksPaper 1~8 min

A researcher measures the heart rate (

y

, in beats per minute) of a person after exercising for

x

minutes. The data for five individuals are:

x

— 2 — 4 — 6 — 8 — 10

y

— 95 — 110 — 125 — 140 — 155 It is given that

\sum x^2 = 220

and

\sum xy = 3900

(a)

State the mean point

(\bar{x},\, \bar{y})

and explain why the regression line of

y

x

must pass through it. [2 marks]

(b)

Find the equation of the regression line of

y

x

in the form

y = a + bx

, giving

a

and

b

correct to three significant figures. [3 marks]

Solutions

3ChallengeSAQ-LRegression analysis8 marksPaper 1~12 min

A biomedical research team investigates the concentration of a drug in a patient's bloodstream. The concentration

y\,\text{mg/L}

is recorded at time

x

hours after administration for six patients.

x

(hours) — 1 — 2 — 3 — 4 — 5 — 6

y

(mg/L) — 8.2 — 6.1 — 4.8 — 3.9 — 3.1 — 2.5 The relationship is modelled by

y = a \times b^x

, where

a

and

b

are constants.

(a)

Show that taking natural logarithms of both sides of

y = a \times b^x

gives a linear model of the form

\ln y = c + dx

, stating

c

and

d

in terms of

a

and

b

. [2 marks]

(b)

Using the data in the table, determine the equation of the regression line of

\ln y

x

. Give the coefficients correct to three significant figures. [3 marks]

(c)

Hence, determine the values of

a

and

b

for the exponential model. Give your answers correct to three significant figures. [3 marks]

Solutions

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