# public Wardje /vub

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mathstat oef - 2.5

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 @@ -387,6 +387,68 @@ \section{Estimation, basic concepts} 387 387   388 388  \section{Hypotheses testing} 389 389   390 +\begin{opgave} 391 +\end{opgave} 392 +\begin{oplossing} 393 +\end{oplossing} 394 + 395 +\begin{opgave} 396 +\end{opgave} 397 +\begin{oplossing} 398 +\end{oplossing} 399 + 400 +\begin{opgave} 401 +\end{opgave} 402 +\begin{oplossing} 403 +\end{oplossing} 404 + 405 +\begin{opgave} 406 +\end{opgave} 407 +\begin{oplossing} 408 +\end{oplossing} 409 + 410 +\begin{opgave} 411 + Suppose a machine makes metal plates of a specific thickness. 412 + In a particular company there are two such devices. 413 + Engineers are worried whether both machines work equally consistent. 414 + To test this, they have 10 plates from machine A and 12 platers from 415 + machine B. 416 + The thickness of each plate is measured and shown here. 417 + $A: 22.3, 21.9, 21.8, 22.4, 22.3, 22.5, 21.6, 22.2, 21.8, 21.6$. 418 + $B: 22.0, 21.7, 22.1, 21.9, 21.8, 22.0, 21.9, 22.1, 22.2, 21.9, 22.0, 22.1$. 419 + 420 + Do both machines work equally consistent (that is: same variance)? 421 + You may suppose that the thickness follows a normal distribution. 422 + $\alpha = 0.05$. 423 +\end{opgave} 424 +\begin{oplossing} 425 + For this we use the F-test of equality of variances, as worked out in the 426 + previous exercise. First off we need the sample mean for each machine. 427 + $\overline{A} = 22.04$, $\overline{B} = 21.975$. Sample variance 428 + then becomes 429 + \begin{align*} 430 + \hat{s}_A^2 &= \frac{\sum (a_i - \overline{A})^2}{n_A-1} = 0.114\\ 431 + \hat{s}_B^2 &= \frac{\sum (b_i - \overline{B})^2}{n_B-1} = 0.0202 432 + \end{align*} 433 + 434 + The test statistic is 435 + $ 436 + T = \frac{\hat{s}_A^2}{\hat{s}_B^2} = \frac{0.114}{0.0202} = 5.64 437 +$ 438 + we use a table to look up the values 439 + \begin{align*} 440 + F_{9;11;0.975} &= 3.588\\ 441 + F_{9;11;0.025} &= 0.256 442 + \end{align*} 443 + It is clear that $5.64 > 3.588$, thus we are in the critical region 444 + and the null hypothesis is rejected. 445 +\end{oplossing} 446 + 447 +\begin{opgave} 448 +\end{opgave} 449 +\begin{oplossing} 450 +\end{oplossing} 451 + 390 452  %\newpage 391 453  %\begin{thebibliography}{99} 392 454  % \bibitem{VOORBEELD} Voorbeeld, Ward Muylaert.