Code
409 | 11 | 22 | 7 | 277 | 726 |
512 | 4 | 14 | 11 | 220 | 761 |
921 | 15 | 36 | 18 | 497 | 1487 |
\[ \def\RR{{\mathbb{R}}} \def\PP{{\mathbb{P}}} \def\EE{{\mathbb{E}}} \def\VV{{\mathbb{V}}} \]
Based on the question:
The null hypothesis is \(H_0: p_{10} = p_{20} = \dots = p_{80} = {1\over 8} = 0.125\).
\(H_0\) is rejected if \(x^2\) value is greater than or equal to the values of \(X^2\), \(\alpha=0.10\), \(df=k-1=7\), \(k=8\).
The chi-squared test with 7 df and \(\alpha=0.10\): \(X^2_{0.10, 7} \approx 12.017\) so \(H_0\) is rejected if \(X^2\geq 12.017\).
\(n=120\). The expected frequency is \(E_i=np_{i0} = 120\cdot 0.125 = 15\).
From the given frequency table, the observed frequencies are: \[ O_1 = 12, O_2 = 16, O_3 = 17, O_4 = 15, O_5 = 13, O_6 = 20, O_7 = 17, O_8 = 10 \]
The test statistic value is \[ \begin{align*} X^2 &= \sum_{i=1}^k{(O_i-E_i)^2\over E_i} = {(12-15)^2\over 15} + \dots + {(10-15)^2\over 15} \\ &= {9\over 15}+{1\over 15}+{4\over 15}+{0\over 15}+{4\over 15}+{25\over 15}+{4\over15}+{25\over 15} \\ &= {24\over 5} = 4.80 \end{align*} \]
Since \(4.80<X^2_{0.10,7}\), the null hypothesis was not rejected.
a)
\[ \begin{align*} \PP(X=x) &= p^{x-1}\overbrace{q}^{1-p}, \quad x=1,2,\dots \\ \mathcal{L}(p) &= \prod_{i=1}^n p^{x_i-1}q = p^{\sum x_i-n}q^n \\ \ell(p) = \log\mathcal{L}(p)&= \left(\sum x_i-n\right)\log p+n\log(1-p) \\ {d\ell(p)\over dp} &= {\sum x_i-n\over p}-{n\over 1-p} \qquad \text{set }=0 \\ 0 &= {\left(\sum x_i-n\right)(1-p)-np\over p(1-p)} \\ 0 &= {\sum x_i - n - p\sum x_i + \color{DarkMagenta}{np - np}\over p(1-p)} \\ p\sum x_i &= \sum x_i-n \\ \hat p &= {\sum x_i-n\over \sum x_i} = {363-130\over 130} \approx .642 \end{align*} \]
b) \(H_0:\) data does not fit the geometric distribution, \(H_a:\) data does fit geometric distribution. MVJ: This is exactly opposite the \(H_0\) and \(H_a\) we established for chi-squared tests.
Observed | E | O-E | \((O-E)^2\) | \((O-E)^2\over E\) | |
---|---|---|---|---|---|
1 | 48 | 46.34 | 1.46 | 2.1516 | 0.04380 |
2 | 31 | 29.87 | 1.13 | 1.2769 | 0.04275 |
3 | 20 | 19.18 | 0.82 | 0.6724 | 0.89002 |
4 | 9 | 12.31 | -3.31 | 10.9561 | 0.45696 |
5 | 6 | 7.90 | -1.90 | 3.6100 | 0.00097 |
6 | 5 | 5.07 | -0.07 | 0.0049 | 0.0009664694 |
\(\geq 7\) | 11 | 3.25 | 7.75 | 60.0625 | 18.48077 |
\(\chi^2 = \sum{(O-E)^2\over E} \approx 20.231\), \(\alpha=0.05\). \(df=k-1-m = 7-1-1 = 5\).
\(\chi^2_{0.05,5}\approx 11.07\)
\(20.231 > 11.07\), reject H_0.
409 | 11 | 22 | 7 | 277 | 726 |
512 | 4 | 14 | 11 | 220 | 761 |
921 | 15 | 36 | 18 | 497 | 1487 |
\(E_i = {\text{row total}\cdot\text{column total}\over\text{total}}\):
449.66 | 7.32 | 17.58 | 8.79 | 242.65 |
471.34 | 7.68 | 18.42 | 9.21 | 254.35 |
\(\chi^2 = \sum\left(O_i-E_i\over \sqrt{E_i}\right)^2 \approx 23.13\)
Assuming \(\alpha=0.05\), with \(4\) df, p-value \(0.0001 < 0.05\).