Home / IBDP Maths analysis and approaches Topic: SL 4.1-Concepts of population, sample, random sample HL Paper 1

# IBDP Maths analysis and approaches Topic: SL 4.1-Concepts of population, sample, random sample HL Paper 1

## Question

At a skiing competition the mean time of the first three skiers is 34.1 seconds. The time for the fourth skier is then recorded and the mean time of the first four skiers is 35.0 seconds. Find the time achieved by the fourth skier.

## Markscheme

total time of first 3 skiers $$= 34.1 \times 3 = 102.3$$     (M1)A1

total time of first 4 skiers $$= 35.0 \times 4 = 140.0$$     A1

time taken by fourth skier $$= 140.0 – 102.3 = 37.7{\text{ (seconds)}}$$     A1 [4 marks]

## Question

The discrete random variable X has the following probability distribution, where p is a constant.

a. Find the value of p.[2]

b.i. Find μ, the expected value of X.[2]

b.ii. Find P(X > μ). [2]

## Markscheme

a.equating sum of probabilities to 1 (p + 0.5 − p + 0.25 + 0.125 + p3 = 1)       M1

p3 = 0.125 = $$\frac{1}{8}$$

p= 0.5      A1

[2 marks]

b.i

μ = 0 × 0.5 + 1 × 0 + 2 × 0.25 + 3 × 0.125 + 4 × 0.125       M1

= 1.375 $$\left( { = \frac{{11}}{8}} \right)$$     A1

[2 marks]

b.ii.

P(X > μ) = P(X = 2) + P(X = 3) + P(X = 4)      (M1)

= 0.5       A1

Note: Do not award follow through A marks in (b)(i) from an incorrect value of p.

Note: Award M marks in both (b)(i) and (b)(ii) provided no negative probabilities, and provided a numerical value for μ has been found.

[2 marks]

### Question

Consider the data $$x_1,x_2,x_3, …, x_n,$$ with mean $$\overline{x}$$, and standard deviation $$s$$.

1. If each number is increased by $$k$$,
1. show that the new mean is $$\overline{x}+k$$ (i.e. it is also increased by $$k$$)
2. show that the new standard deviation is $$s$$(i.e. it remains the same)
2. If each number is multiplied by $$k$$
1. show that the new mean is $$k\overline{x}$$ (i.e. it is also multiplied by $$k$$)
2. show that the new standard deviation is $$ks$$(i.e. it is also multiplied by $$k$$)
3. write down the relation between the original and the new variance.

Ans:

1. $$\overline{x}_{new}=\frac{\sum_{i=1}^{n}(x_i+k)}{n}=\frac{\sum_{i=1}^{n}x_i+\sum_{i=1}^{n}k}{n}=\frac{\sum_{i=1}^{n}x_i}{n}+\frac{\sum_{i=1}^{n}k}{n}=\overline{x}+\frac{kn}{n}=\overline{x}+k$$
$$s_{new}=\sqrt{\frac{\sum_{i=1}^{n} ((x_i+k)-(\overline{x}-k))^2 }{n}}=\sqrt{\frac{\sum_{i=1}^{n}(x_i-\overline{x})^2)}{n}}=s$$
2. $$\overline{x}_{new}=\frac{\sum_{i=1}^{n}(kx_i)}{n}=\frac{k\sum_{i=1}^{n}(x_i)}{n}=k\frac{\sum_{i=1}^{n}(x_i)}{n}=k\overline{x}$$
$$s_{new}=\sqrt{\frac{\sum_{i=1}^{n}(kx_i-k\overline{x})^2}{n}}=\sqrt{\frac{k^2\sum_{i=1}^{n}(x_i-\overline{x})^2}{n}}=k\sqrt{\frac{\sum_{i=1}^{n}(x_i-\overline{x})^2}{n}}=ks$$
$$s_{new}=ks$$⇒$$s^2_{new}=k^2s^2$$
so the original variance is multiplied by $$k^2$$.

### Question

Consider the following data

 x 1 2 3 4 y 2 3 7 8
1. Find the mean and the variance for the values of $$x$$.
2. Find the mean and the variance for the values of $$y$$.
3. Find the correlation coefficient $$r$$.
4. Describe the relation between $$x$$ and $$y$$.
5. Find the equation $$y = ax+b$$ of the regression line for $$y$$ on $$x$$.
6. Find the equation $$x = cy+d$$ of the regression line for $$x$$ on $$y$$.
7. Find the inverse of the function in question $$(e);$$ Is it the function in question $$(f)$$?
1. $$\mu_x = 2.5, \sigma_x^2= 1.11803^2= 1.25$$
2. $$\mu_y = 5, \sigma_y^2=2.54951^2= 6.5$$
5. $$y = 2.2x – 0.5$$
6. $$x = 0.423y + 0.385$$
7. $$y = 2.2x – 0.5$$ ⇔ $$y + 0.5 = 2.2x$$ ⇔ $$x = 0.455 y + 0.227$$. They are different