Maths - Statistics Y2

2 possible log equations for exponential models

1. y = ax^b

2. y = ab^x

How do you measure the strength and type of linear correlation?

The product moment coefficient

What must you check when using the inverse normal function?

It is in the format P(Z<z) = …

It will NOT work if in the format P(Z>z) = …

What values can PMCC take?

Values between -1 and 1

What does the PMCC describe?

The linear correlation between 2 variables

What is the symbol for the PMCC of a sample?

r

What does it mean if r = 1?

There is perfect positive linear correlation

What does it mean if r = -1?

There is perfect negative linear correlation

What does it mean if r = 0?

There is no linear correlation

What is the symbol for the PMCC of a population?

ρ

When carrying out a hypothesis test for the PMCC, when do you use a one tailed test?

To test if there's a positive / negative correlation

When carrying out a hypothesis test for the PMCC, when do you use a two tailed test?

To test if there's a correlation

A ∪ B meaning

A ∩ B meaning

A’ meaning

(i.e. everything but A)

If A and B are mutually exclusive, what does P(A ∪ B) = ?

P(A) + P(B)

When does P(A ∪ B) = P(A) + P(B)?

When A and B are mutually exclusive

If A and B are independent, what does P(A ∩ B) = ?

P(A) x P(B)

When does P(A ∩ B) = P(A) x P(B)?

When A and B are indepedent

Whole sample space symbol

ℰ

P(B|A)

The probability of B occurs given that A has already occurred

When A and B are independent, what does P(A|B) = ?

P(A|B’) and P(A)

The addition formula for 2 events, A aand B

P(A ∪ B) = P(A) + P(B) − P(A ∩ B)

The multiplication formula for 2 events, A and B, to find P(B|A)

P(B|A) = P(B ∩ A) / P(A)
P(B ∩ A) = P(B|A) × P(A)

Continuous random variable

A variable that can take on one of infinitely many values

In a normal distribution, what is the probability that a continuous random variable takes any one specific value?

0

What is the area under a continuous probability distribution equal to?

1

5 features of the normal distribution curve

1. Has parameters μ for mean and σ² for variance

2. Symmetrical - mean = mode = median

3. Bell-shaped curve with asymptotes at each end

4. points of inflection at ‘μ + σ’ and ‘μ - σ’

5. Total area under curve = 1

What is the significance of the normal distribution curve being symmetrical?

The mean = mode = median

How is a normal distribution represented?

X ~ N(μ,σ²)

For a normal distribution where you know p, how do you find the value of a in P(x<a) = p?

By using the inverse normal distribution

In a normal distribution, what proportion of data lie below Q1 and below Q3?

Below Q1 = 25%

Below Q3 = 75%

What is the mean and standard deviation of the standard normal distribution?

Mean = 0

Standard deviation = 1

How do you code the normal distribution to be the standard normal distribution?

How is the standard normal distribution represented?

Z ~ N(μ,σ²) or Φ(a)

How do you find the mean or standard deviation of a normal distribution from the probability/area?

1. Code the values to be the standard normal distribution

2. Use the inverse normal distribution and solve

When can you approximate a binomial distribution to a normal distribution? (2)

1. If n is large

2. If p is close to 0.5

How do you approximate a binomial distribution to a normal distribution?

X ~ B(n,p) is approximately equal to N(μ,σ²) where

μ = np

σ = root of [np(1-p)] [so σ² = np(1-p)]

What do you need to do when approximating a binomial distribution to a normal distribution?

Apply a continuity correction

When testing a hypotheses about the mean of a normally-distributed random variable, what is the distribution used?

When testing a hypothesis about the mean of a normally-distributed random variable, how do you find the critical region / value?

By using the inverse normal distribution or by standardising the test statistic then using the table

How do you carry out a hypothesis test for the mean of a normally-distributed random variable?

1. Write out H₀ and H₁

2. Assume H₀ and write out the distribution

3. Write out the sample mean distribution

4. Find the critical value, x, when P(X<x) = a, where a is the level of significance for one-tailed / half the level of significance for two-tailed

5. Find the sample mean

When approximating a normal distribution from a binomial distribution, what would you calculate when:

i. X >/= 5

ii. X </= 6

iii. 5 </= X </= 6

i. Y > 4.5 (go for 0.5 below actual)

ii. Y < 6.5 (go for 0.5 above actual)

iii. 4.5 < Y < 6.5

After carrying out a hypothesis test on a normal distribution, how do you find the p-value?

Multiply your calculated value by 2

Why might a curve not be suitable for a normal distribution?

It is not symmetrical / bell-shaped

When testing a hypothesis about the mean of a normally-distributed random variable, how do you decide where the inequality sign faces?

If test statistic is greater than mean of H₀, then >

If test statistic is less than mean of H₀, then <

How do you carry out a hypothesis test for the PMCC? (4)

1. Write out H₀ and H₁

2. Calculate r for the data

3. Find critical value using the significance level of the test and the sample size

4. If testing for negative correlation and r is less than critical value: reject H₀

   If testing for positive correlation and r is more than critical value: accept H₀

When adding new values to a data set closer to the mean than one standard deviation, what would happen to the standard deviation and why?

It would decrease because of the new values being closer than one standard deviation to the mean

How do you calculate P(A|B)?

Probability of A∩B / P(B)

When might a normal distribution not be a good model?

If it suggests that there is a non-negligible value for the probability of the variable being < 0 (in the cases where this is impossible e.g. time)

How do you calculate the percentage error when approximating a binomial distribution?

(binomial - normal approximation)/binomial

2 variables on LDS that would not be suitable for being modelled with a normal distribution

1. Daily Mean Windspeed (Beaufort) - qualitative data

2. Rainfall - not symmetric