Statistics 2 Formula AQA

```Stats 2 (AQA) Formula
(everything in blue in formula book)
Expectation (mean) &amp; Variance of Random Variable from Probability Distribution
πππ(π) = π 2 = ∑ π₯π 2 ππ − π 2
πΈ(π) = π = ∑ π₯π ππ
= πΈ(π 2 ) − π 2
( ο­ οΉ x . ο­ is theoretical mean, x is sample mean.)
πππ(ππ + π) = πππ(ππ) = π2 πππ(π)
πΈ(ππ + π) = πΈ(ππ) + π = ππΈ(π) + π
Poisson distribution (a discrete distribution)
π(π = π₯) = π −π
π~ππ(π)
ππ₯ =
ππ₯
π₯!
π = π, π 2 = π
π
π
π₯ π₯−1
If π~ππ(π1 ) and π~ππ(π2 ), then the distribution of (π + π)~ππ(π1 + π2 ).
Continuous Random Variables
Probability Density Function (pdf) = f
Cumulative Distribution Function (cdf) = F
π
π(π₯) = πΉ(π₯)
ππ₯
πΉ(π₯) = ∫ π(π‘) ππ‘
π₯
−∞
(replace lower limit with lower limit of function when calculating)
πΉ(−∞) = 0
πΉ(+∞) = 1
Three methods to find specific probabilities of X:
π
1. π(π &lt; π &lt; π) = ∫π π(π₯) ππ₯ (ie integrate to find area between two values)
2. Having integrated, evaluate between limits (similar to finding normal distribution
probabilities).
π(π &lt; π &lt; π) = πΉ(π) − πΉ(π)
πΉ(π &gt; π) = 1 − πΉ(π)
3. Use geometry of graph of π(π₯) (where possible, splitting it into triangles, trapezia etc).
To find median value, m, solve for m either:
π
or
0.5 = ∫−∞ π(π₯)ππ₯
0.5 = πΉ(π)
To find, for example, the value at 95th percentile, solve for d;
π
or
0.95 = ∫−∞ π(π₯)ππ₯
Mean and variance…
0.95 = πΉ(π)
∞
πΈ(π) = π = ∫ π₯π(π₯) ππ₯
−∞
∞
2
πΈ(π 2 ) = ∫ π₯ π(π₯) ππ₯
−∞
∞
πΈ(π(π)) = ∫ π(π₯)π(π₯) ππ₯
−∞
If distribution is symmetrical then πΈ(π) = ππππ‘πππ π£πππ’π.
∞
πππ(π) = πΈ(π 2 ) − π 2 = ∫ π₯ 2 π(π₯) ππ₯ − π 2
−∞
Rectangular (Uniform) Distribution
π(π₯) =
1
πΈ(π) = (π + π)
2
1
π−π
πππ(π) =
πΉ(π₯) =
1
(π − π)2
12
π₯−π
π−π
Estimation
The t distribution (two random variables, πΜ and π, i.e. unknown population variance).
π=
πΜ − π
π
√π
π~π‘π or π~π‘π−1
π = ππ’ππππ ππ πππππππ  ππ πππππππ = π − 1
(parameter π pronounced ‘nu’)
Confidence intervals given by
π₯Μ &plusmn; &quot;c&quot;
π
√π
π 2
π₯Μ &plusmn; &quot;c&quot;√
π
Hypothesis Testing
Null Hypothesis
π»0 : π = 435
Alternative Hypothesis
π»1 : π &lt; 435
One tailed
π»1 : π ≠ 435
Two tailed
Acceptance of null hypothesis does not mean it is true, rather that the data provides no
evidence to prefer the alternative.
Type 1 error – to reject H0 (and accept H1) when H0 is actually true.
Type 2 error – to reject H1 (and accept H0) when H1 is actually true.
π(ππ¦ππ 1 πππππ) = π πππππππππππ πππ£ππ
Test Procedure:
1. Write down the two hypotheses.
2. Identify an appropriate test statistic and the distribution of the corresponding random
variable.
3. Identify the significance level (usually given). This is also π(ππ¦ππ 1 πππππ).
4. Determine the critical region (should be done before collecting data).
5. Calculate the value of the test statistic.
6. Determine and clarify in context the outcome of the test.
ππ Contingency Tables Tests
Part A – The ππ Distribution
π 2 ~ππ2
π 2 = 2π
(where π = π − 1)
π=π
π
(ππ − πΈπ )2
π =∑
πΈπ
2
π=1
(where π = number of different possible outcomes)
All expected frequencies must be greater than 5
Part B – Contingency Tables
Associated vs independent
To calculate expected frequencies from observed frequencies use
πππ€ π‘ππ‘ππ &times; ππππ’ππ π‘ππ‘ππ
πππππ π‘ππ‘ππ
π = (π − 1)(π − 1)
Yates’s correction for a 2x2 table (π = 1 ):
4
ππΆ2
=∑
1
(|ππΌ − πΈπ | − 0.5)2
πΈπ
π
π 2
=
(|ππ − ππ| − )
ππππ
2
Where…
a
c
r
b
d
s
m
n
N
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