Hypothesis Testing: Comparing Two Population Means

Hypothesis Testing for 2 Population Means

• Objective is to compare the means of two populations.

Case 1: Known Population Standard Deviations ($\sigma_1, \sigma_2$)

• Null hypothesis: $H_0: \mu_1 = \mu_2$
• Alternative hypothesis: $H_1: \mu_1 \neq \mu_2$ or $\mu_1 > \mu_2$ or $\mu_1 < \mu_2$
• Test statistic: $Z = \frac{(\bar{x}_1 - \bar{x}_2) - (\mu_1 - \mu_2)}{\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}}$

Case 2: Unknown but Equal Population Standard Deviations ($\sigma_1 = \sigma_2 = \sigma$)

• Null hypothesis: $H_0: \mu_1 = \mu_2$
• Alternative hypothesis: $H_1: \mu_1 \neq \mu_2$ or $\mu_1 > \mu_2$ or $\mu_1 < \mu_2$
• Test statistic: $T = \frac{(\bar{x}_1 - \bar{x}_2) - (\mu_1 - \mu_2)}{S_p \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}}$
• $S_p^2 = \frac{(n_1 - 1)S_1^2 + (n_2 - 1)S_2^2}{n_1 + n_2 - 2}$ is the pooled variance estimator.
• Degrees of freedom: $n_1 + n_2 - 2$

Case 3: Unknown and Unequal Population Standard Deviations ($\sigma_1 \neq \sigma_2$)

• Null hypothesis: $H_0: \mu_1 = \mu_2$
• Alternative hypothesis: $H_1: \mu_1 \neq \mu_2$ or $\mu_1 > \mu_2$ or $\mu_1 < \mu_2$
• Test statistic: $T = \frac{(\bar{x}_1 - \bar{x}_2) - (\mu_1 - \mu_2)}{\sqrt{\frac{S_1^2}{n_1} + \frac{S_2^2}{n_2}}}$
• Degrees of freedom: $d.f. = \frac{(\frac{S_1^2}{n_1} + \frac{S_2^2}{n_2})^2}{\frac{(\frac{S_1^2}{n_1})^2}{n_1 - 1} + \frac{(\frac{S_2^2}{n_2})^2}{n_2 - 1}}$

Matched Pair Hypothesis Test

• Examines the mean difference between paired observations
• $d_i = x_{1i} - x_{2i}$
• Null hypothesis: $H_0: \mu_d = 0$
• Alternative hypothesis: $H_1: \mu_d \neq 0$ or $\mu_d > 0$ or $\mu_d < 0$
• Test statistic: $T = \frac{\bar{d} - \mu_d}{S_d / \sqrt{n}}$
• Degrees of freedom: $n - 1$

Hypothesis Testing for 2 Population Proportions

• Aims to compare the proportions of two populations
• Null hypothesis: $H_0: p_1 = p_2$
• Alternative hypothesis: $H_1: p_1 \neq p_2$ or $p_1 > p_2$ or $p_1 < p_2$
• Test statistic: $Z = \frac{(\hat{p}_1 - \hat{p}_2) - (p_1 - p_2)}{\sqrt{\hat{p}(1 - \hat{p})(\frac{1}{n_1} + \frac{1}{n_2})}}$
• $\hat{p} = \frac{x_1 + x_2}{n_1 + n_2}$

Fourier Series

Periodic Functions

• A function $f(x)$ is periodic if $f(x + T) = f(x)$.
• $T$ represents the period of $f$.
• Examples of periodic functions include $\sin x$, $\cos x$ (period $2\pi$), and $\tan x$ (period $\pi$).

Fourier Series Representation

• Used for periodic functions with period $2L$
• $f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} \left( a_n \cos \frac{n \pi x}{L} + b_n \sin \frac{n \pi x}{L} \right)$
• Fourier coefficients are:
  • $a_0 = \frac{1}{L} \int_{-L}^{L} f(x) , dx$
  • $a_n = \frac{1}{L} \int_{-L}^{L} f(x) \cos \frac{n \pi x}{L} , dx$
  • $b_n = \frac{1}{L} \int_{-L}^{L} f(x) \sin \frac{n \pi x}{L} , dx$

Convergence of Fourier Series

• Relies on Dirichlet Conditions
  • $f(x)$ is defined on the interval $(-L, L)$.
  • $f(x)$ is piecewise continuous on the interval $(-L, L)$.
  • $f(x)$ has a finite number of maxima and minima on the interval $(-L, L)$.
• Converges to $f(x)$ at points where $f$ is continuous.
• Converges to $\frac{f(x^+) + f(x^-)}{2}$ at points of discontinuity.

Example: Square Wave

• $f(x) = \begin{cases} -1, & -\pi < x < 0 \ 1, & 0 < x < \pi \end{cases}$
• $f(x + 2\pi) = f(x)$
• Fourier Series Representation: $f(x) = \frac{4}{\pi} \sum_{k=0}^{\infty} \frac{\sin((2k+1)x)}{2k+1}$

Properties of Fourier Series

• Linearity: Coefficients scale linearly with function scaling ($a_n, b_n$ become $\alpha a_n + \beta c_n$ and $\alpha b_n + \beta d_n$ for $\alpha f(x) + \beta g(x)$)
• Time Shifting: Shifted coefficients are computed using trigonometry identities
  • $a_n' = a_n \cos \frac{n \pi x_0}{L} + b_n \sin \frac{n \pi x_0}{L}$
  • $b_n' = -a_n \sin \frac{n \pi x_0}{L} + b_n \cos \frac{n \pi x_0}{L}$
• Time Scaling: $f(ax)$ has period $\frac{2L}{a}$
• Differentiation: Term-by-term differentiation yields $f'(x)$ if conditions are met.
• Integration: Term-by-term integration to get $\int f(x) , dx$.

Parseval's Theorem

• Relates the energy of $f(x)$ to its Fourier coefficients
• $\frac{1}{2L} \int_{-L}^{L} |f(x)|^2 , dx = \frac{|a_0|^2}{4} + \frac{1}{2} \sum_{n=1}^{\infty} (|a_n|^2 + |b_n|^2)$

Gibbs Phenomenon

• At jump discontinuities, overshoot occurs, not disappearing with more terms, reaching ~9% of the jump size.