Further Statistics 2

What is the equation of the least squares regression line of y on x?

y = a + bx

b = Sxy/Sxx

a = ȳ − bx̄

Sxy = Σxy − (ΣxΣy)/n

Sxx = Σx² − (Σx)²/n

How do you interpret a and b in a regression line?

b: for every 1 unit increase in x, y increases/decreases by b units

a: predicted value of y when x = 0

What is a residual and what is the residual sum of squares (RSS)?

Residual = actual y − predicted y

Sum of all residuals = 0

RSS = Syy − b·Sxy

Smaller RSS = better fit

Only compare RSS if datasets have same units

How do you assess whether a linear model is suitable using residuals?

Plot residuals against fitted values

Random scatter around 0 → linear model suitable

Obvious curve or pattern → linear model not suitable

How do you handle coded data in regression?

Sub the coding substitutions algebraically into the regression line

Rearrange to get original regression line in terms of original variables

What is the difference between interpolation and extrapolation?

Interpolation: predicting within the range of data → reliable

Extrapolation: predicting outside the range → unreliable, should not be used

What is the PMCC and when is it appropriate?

r = Sxy / √(Sxx·Syy)

Range: −1 to 1

Use when: data is continuous, bivariate normal distribution, testing linear correlation

Coding does not affect PMCC

What is Spearman’s rank correlation coefficient and when do you use it instead of PMCC?

rs = 1 − (6Σd²)/(n(n²−1))

d = difference in ranks for each pair

Use instead of PMCC when: data not continuous, not normal, already ranked, or no linear correlation apparent

How do you handle tied ranks?

Assign each tied value the mean of the ranks they would have occupied | If many ties: use PMCC instead

What are the steps for a hypothesis test for zero correlation?

1. H₀: ρ=0, H₁: ρ≠0 (two-tailed) or ρ>0/ρ<0 (one-tailed)

2. Calculate r or rs

3. Look up critical value from table

4. If |observed| > critical → reject H₀ → evidence of correlation

5. Conclude in context

What are the conditions for a valid probability density function (pdf)?

f(x) ≥ 0 for all x

∫f(x)dx = 1 over full range

P(a<X<b) = ∫f(x)dx from a to b

P(X=a) = 0

What is the cumulative distribution function (cdf) and how does it relate to the pdf?

F(x) = P(X≤x) = ∫f(t)dt

f(x) = F’(x) (differentiate cdf to get pdf)

F(lower bound) = 0, F(upper bound) = 1

How do you find the constant k in a piecewise pdf?

Set ∫f(x)dx = 1 over full range

Integrate, substitute limits, solve for k

How do you find the constant c in a piecewise cdf?

At lower boundary: F(x) = 0

At upper boundary: F(x) = 1

Ensure continuity at boundaries of each piece

How do you find E(X) and Var(X) for a continuous distribution?

E(X) = ∫x·f(x)dx

Var(X) = E(X²) − [E(X)]² where E(X²) = ∫x²·f(x)dx

For piecewise: integrate each piece separately then add

E(aX+b) = aE(X)+b | Var(aX+b) = a²Var(X)

How do you find the median, quartiles and percentiles?

Median: F(m) = 0.5

Lower quartile: F(Q₁) = 0.25

Upper quartile: F(Q₃) = 0.75

nth percentile: F(Pn) = n/100

Set F(x) equal to target value and solve

How do you find the mode of a continuous distribution?

Differentiate f(x), set f’(x) = 0, solve for x

Check it is a maximum

Shortcut: if f(x) strictly increasing → mode = upper boundary; strictly decreasing → mode = lower boundary

How do you determine skewness?

Positive skew: mode < median < mean

Negative skew: mean < median < mode

Symmetric: mean = median = mode

What is the continuous uniform distribution X~U[a,b]? And its mean and variance?

f(x) = 1/(b−a) for a≤x≤b

E(X) = (a+b)/2

Var(X) = (b−a)²/12 | F(x) = (x−a)/(b−a)

Rounding errors follow U(−0.5, 0.5)

What are the rules for mean and variance when combining random variables?

E(aX+bY) = aE(X)+bE(Y)

E(aX−bY) = aE(X)−bE(Y)

Var(aX+bY) = a²Var(X)+b²Var(Y)

Var(aX−bY) = a²Var(X)+b²Var(Y)

Variances always ADD regardless of + or −

What is the distribution of a linear combination of normal variables?

If X~N(μ₁,σ₁²) and Y~N(μ₂,σ₂²) independently:

aX+bY ~ N(aμ₁+bμ₂, a²σ₁²+b²σ₂²)

aX−bY ~ N(aμ₁−bμ₂, a²σ₁²+b²σ₂²)

What is the distribution of the sum of n identical independent normal variables?

If X₁,…,Xₙ ~ N(μ,σ²) independently: ΣXᵢ ~ N(nμ, nσ²)

Note: adding n copies → variance ×n | Multiplying one variable by n → variance ×n²

How do you find P(X > Y) for two independent normal variables?

1. Define C = X−Y

2. Find distribution of C using combination rules

3. Find P(C>0) using normal calculator

What is an unbiased estimator?

A statistic T where E(T) = θ (the true parameter)

Bias = E(T) − θ | Unbiased → bias = 0

Unbiased estimate of μ: x̄

Unbiased estimate of σ²: s² = Σ(xᵢ−x̄)²/(n−1)

What is the standard error of the mean?

SE = σ/√n

If σ unknown: use s/√n

Decreases as n increases

What is a confidence interval for the population mean?

CI = x̄ ± z·(σ/√n) |

z values: 90%→1.6449, 95%→1.96, 99%→2.5758 |

Width = 2z·(σ/√n)

How do you find the minimum sample size n for a target CI width?

Set 2z·(σ/√n) ≤ target width

Solve for n

Round up to nearest integer

How do you find the confidence level from a given interval?

Width = upper − lower | z = width/(2·σ/√n)

Find P(−z < Z < z) from tables

Multiply by 100 for percentage

What are the steps for a hypothesis test for a population mean (σ known)?

1. H₀: μ=μ₀, state H₁

2. Calculate Z = (x̄−μ₀)/(σ/√n)

3. Look up critical z valu

4. |Z|>critical → reject H₀ |

5. Conclude in context

6. Use CLT if population not normal but n large

How do you test for the difference between two means?

H₀: μx=μy |

Test statistic: Z = (x̄−ȳ)/√(σx²/nx + σy²/ny)

Compare to critical z

Two-tailed if H₁: μx≠μy, one-tailed otherwise

How do you find unbiased estimates from raw data?

x̄ = Σx/n

s² = (Σx² − (Σx)²/n)/(n−1)

For combined samples: combine Σx and Σx² first, then recalculate

What is the confidence interval for a population variance?

[(n−1)s²/χ²upper , (n−1)s²/χ²lower]

Degrees of freedom ν = n−1

For 95% CI: use χ²(0.025) and χ²(0.975)

Assumption: sample from normal distribution

What are the steps for a chi-squared test for variance?

1. H₀: σ²=k, H₁: σ²k or ≠k

2. ν = n−1 |

3. Test statistic: (n−1)s²/σ²

4. Compare to critical χ² value

5. One-tailed lower: reject if χ²upper critical | Two-tailed: reject if χ²upper

What is the F-distribution and when is it used?

Used to test whether two normal populations have equal variances

F = S²larger/S²smaller

Parameters: ν₁=larger sample df, ν₂=smaller sample df

Written F~F(ν₁,ν₂)

How do you find a lower critical value for the F-distribution?

Lower critical value = 1/F(ν₂,ν₁)(p) — swap the degrees of freedom and take reciprocal

What are the steps for an F-test for equal variances?

1. Find s²larger and s²smaller

2. ν₁=larger n−1, ν₂=smaller n−1

3. H₀: σ₁²=σ₂²

4. Look up F(ν₁,ν₂) — for two-tailed halve significance level

5. F_test = s²larger/s²smaller

6. If F_test > critical → reject H₀

7. Assumption: data normally distributed

When do you use the t-distribution instead of normal?

When σ is unknown, population is normal, and sample is small

t = (x̄−μ)/(s/√n)

Degrees of freedom ν = n−1

As ν→∞, t-distribution → standard normal

What is the confidence interval for a mean using the t-distribution?

x̄ ± t(ν,α/2) × s/√n | ν = n−1

Look up t(ν)(0.025) for 95% CI

Assumption: population is normally distributed

What are the steps for a t-test for a population mean?

1. State H₀: μ=μ₀, H₁

2. Calculate t = (x̄−μ₀)/(s/√n)

3. ν = n−1, look up critical t

4. |t|>critical → reject H₀

5. Conclude in context

When do you use a paired t-test? And what are the steps?

Use when: two sets of results on the same subjects (before/after)

1. Calculate differences d

2. H₀: μD=0

3. t = (d̄−0)/(sD/√n), ν=n−1

4. Compare to critical t

Assumption: differences are normally distributed

What is the pooled variance estimate and when is it used?

Use when: two independent samples, σ unknown but assumed equal

sp² = [(nx−1)sx² + (ny−1)sy²] / (nx+ny−2)

Degrees of freedom: ν = nx+ny−2

What are the steps for a two-sample t-test with equal unknown variances?

1. H₀: μx=μy

2. Calculate x̄, ȳ, sx², sy², sp²

3. t = (x̄−ȳ)/(sp√(1/nx+1/ny))

4. 4. ν = nx+ny−2, look up critical t

5. Compare and conclude

Assumptions: independent samples, equal variances, both populations normal

What is the confidence interval for the difference between two means (equal unknown variances)?

(x̄−ȳ) ± t(ν,α/2) × sp√(1/nx+1/ny)

ν = nx+ny−2