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Numerical Answers (workings/explanations are after the numerical answers.)

Question 1:

Question 2:

Question 3:

Question 4:

Question 5:

Question 6:

Question 7:

Question 8: ; orange bird & yellow bird, orange bird & white rider, white horse & white rider, white horse & yellow bird

Question 9:

Question 10:

Question 11:

(i)

(ii)

(iii)

, a constant.

(i)

(ii)

From the graph,

(iii)

let

Since , V is maximum when

(i)

When ,

(ii)

(ii)

(iv)

We observe that the answers found in (ii) and (iii) are accurate up to 5 decimal places. As the maclaurin’s series is centred about and the value used in (ii) and (iii) is close to zero, the approximation is good and accurate.

(v)

We observe that is not defined at . Thus, the maclaurin’s series cannot be used.

(i)

At ,

(ii)

Since and are collinear,

(i)

These 22 clubs form a sample, as they are observations drawn from the population of 100 clubs.

(ii)

Dilip can collect a random sample by listing the 100 clubs in order and picking a random number from 1 to 4. For example, he picks the the number 4 and will then sample every fourth club in the list. This will ensure that each club has an equal chance of being selected, independently of each other.

(iii)

Required number of ways ways.

(i)

The probability that a randomly chosen mug is faulty is constant throughout.

The event that a randomly chosen mug is faulty is independent of another randomly chosen mug being faulty.

(ii)

(iii)

Let be the number of days where at least 7 faulty mugs are found.

(iv)

Required probability

(v)

Required probability

(i)

Required probability

(ii)

Required probability

(iiia)

Required probability

(iiib)

Required probability

(iii)

Let and be number of characters that are Gerri’s two favourites.

We need the number of characters to give a product of 20. Thus the possible combinations are:

Orange Bird & Yellow Bird,

Orange Bird & White Rider,

White Horse & White Rider,

White Horse & Yellow Bird

(i)

Manager should carry out a 2-tail test since he wants to find out if the resistance differs from 750 ohms.

Let denote the population mean resistance of the resistor.

Let and denote the null hypothesis and alternative hypothesis respectively.

(ii)

Sample mean

Under

Test statistics,

Using the GC to perform a 2-tailed test, we have that

Since the , we do not reject at level of significance and conclude with insufficient evidence that the mean resistance is not 750 ohms.

(iii)

As the distribution is unknown, a sample size of 30 will be required in order to approximate the sample mean distribution to a normal distribution by Central Limit Theorem. The null hypothesis will be instead.

(i)

(a),(b)

Note that the coordinates (45, 20) should be on the line. Thus no residuals.

(c)

Using GC, sum of the squares of the residuals

(d)

The residuals might result in negative values, thus when finding sum of the residuals, we might have values cancelling each other out. By squaring the residuals, we ensure the values will always be positive.

(ii)

Bhani’s model will give a better fit since it has a smaller value for the sum of the squares of the residuals.

(iii)

The line must pass through the sample mean of the bivariate data, i.e.,

(iv)

Product moment correlation coefficient

(v)

The estimate is not reliable as it does not lie within the given data range, thus we are extrapolating.

(vi)

Her data points are a perfectly linear and lies on her line.

(i)

Let and denote the mass of a white and a black ball respectively.

(ii)

(iii)

(iv)