s1 mistakes + theory [IAL]

Find the mean and standard deviation from this stem and leaf diagram (4marks)

for stem and leaf, use that formula of S.D.

s.d = √451 = 21.2

Give 2 advantages and disadvantages of using statistical models

advantage:
- it helps make predictions
- they are quick and easy to make/produce
(they can simplify a complext situtation)

disadvantage:
- simplifying a model may result in error as they only give partial description of the real situation
- the model may only work in certain conditions


(NOTE: a mathmetical model is a simplification of a real world situation)

steps for designing a statistical model

(a) interpret the (gradient of) regression line/ =
for every increase in x by 1(unit), y increases by gradient (unit)

(b) easy, got it right.

(c) ALWAYS CHECK IF VALUE INSIDE OR OUTISDE THE RANGE:
value inside range → reliable, outside range —> unreliable

-scatter diagram shows the points lie close to a straight line

- PMCC/ r is close to 1

(therefore a a linear regression model is suitable/supported)

just for concept:
do these sums carefully, easy but lengthy algebra

P.S → always include plugging in value to quadratic formula, or else -1.

just show that P(C) + P(D) > 1

by taking 1 as common, so its in form: 1 + x……

Finding median and mean from histogram

btw +- is does not have any effect on var (only multiply and divide has effect)

Give two other reasons why statistical models are used (2 marks)

• To help understand real world situations

• To simplify complex situations to make predictions
  (To make predictions)

𝑦 = 𝑎 + 𝑏𝑥

d. Give an interpretation of the value of 𝑎
e. Discuss the reliability of using this model to predict her household's energy consumption in the summer.

d. The estimated daily energy consumption( y ) when the average daily temperature( x ) is 0°C

(the estimated value of y when x is 0)

e. The model may not be reliable because the data is based on winter temperatures, and extrapolating to summer temperatures may not be accurate (as temp in summer is much higher)

given n = 32, find median

since n = 32, which is even number.

THEREFORE, median is n/2 = 32/2 = 16th value.
so median = 16th + 17th / 2

Answer 2(b)

Ans: As the population/t increases, GDP/g increases (since r is close to 1 & strong +ve corelation)

(Examiner report mentions:
Candidates responses should clearly state the variables when they describe the relationship between them. Many failed to contextualise their answer by merely stating ‘positive correlation’ or ‘strong correlation’. )

for these questions, interpret, state the FINAL OUTCOME.

If question says:

• “increase”, “change”, “difference”
  👉(y = a +bx), IGNORE a, use only b

NOTE:

P (A ∩ B)
P (A | B ) = —————
P (B)

the reason we keep multiplying for (e) is because the game goes J P J ….. (given J starts)

FULL theory

(a) discrete Uniform distribution

(b) Tossing/rolling a fair die
(where if X is the outcome of the die then X follows discrete Uniform distribution)

theory of mean, median and S.D from table

(ii) mean would decrease as total exercise time for the 0 \leq t < 15 group is reduced,
thus lowering the overall average

(iii) S.D increases because the new info gives more variability at the lower end of the data
(cuz 18 peoples time is reduced to 0 mins)

NOTE → since both sides have the same function (Φ), you can cancel them out

Game is continous like → A B C D A B C D .. until someone wins, thus you keep multiplying

hence the powers are so large, like in (ii) → [ 5/6 × 5/6 x 5/6 x 5/6 x 5/6 x 5/6 x 5/6 = (5/6)^7 ]

mean ≈ median → Symmetrically distributed

(also as yk right tail → greater positive skew, left tail greater → negative skew

NOTE → it doesnt explicitly say the plants are returned or smt, so we consider not returned, hence its 408 and then 407

NOTE:
READ THE Q., it says GIVEN….

GIVEN —> P ( A | B)

INDEPENDENT -→ means in this context, first throw doesn’t affect second throw’s probability

READ QUESTION CAREFULLY, here it says variance of 6, so convert to S.D

s.d = √var = √36 = 6

N ~ ( u, 6²)
[i used 36 instead of 6 the whole time thus got it wrong]

How far data points are from the mean on average, or how spread out

if new value = mean, S.D decreases

if new value < OR > than mean, you gotta check → are the data points are less spread or more spread now
(more spread = more S.D and vice versa)

just simple unitary method

easy, but just be careful when changing from > to < , make sure to keep in mind 1- fi, and using big table

Mean = Median

so decision is justified, as normal distribution is suitable due to symmetry

(i.e the rule of game was difference between red and blue die, so plz read q. carefully, it gives hints)

in statistics →range is the difference between max and min

for e.g if it was 500 kg to nearest 30kg, it would be 30/2 = 15

so it would be: 485 < W < 515

(a)

- points will lie close to a straight line

- there is linear relationship and a negative gradient

[ y intercept doesn’t change even if gradient doubles cuz:
y = mx + c
when m = 0 → y = c

ALSO for:
y = 67mx + c
when m = 0 → y = c

in summary:
new median i found is greater than old, hence median increases

new mean is greater as more higher values present, so sum of fx greater, hence mean greater

give → mean 10 and standard deviation

i almost got all marks, basically when x² -3x -40 > 0
X> 8 OR X< -5

so p( X >8 ) + p(X< -5)

(+ sign as it is OR)
i multplied but should have added

again pretty easy, just gotta be a bit more careful

given outlier is
more than 1.5(IQR)
less than 1.5(IQR)

here since Q1 is changed, you gotta find new outliers

( i forgot to find the new outliers……..)


if Q1 increases from 24 →26, the numbers added must be more than Q1 (but less than median/Q2 as Q2 is same in question)

also since Q2 and Q3 is same, the other number must be more than Q3 to balance out this change (also cant be more than 45 as 45 is UL)


SO answer for (e) So one between 26 and 30 and one between 34 to 45

since its given area in 10m²,

x = 800 → x = 80 m²

no change as we found x = 90 (x = 900 but in 10m² its 90)

mean x = 90

(x -x}(y-y} = 0
so new Sxy = old sxy + 0

thus Sxy remains the same