Question

4. Water samples of 100 cc each are taken periodically from a waste water plant run off. The density of the harmful lime is the parameter of interest. Suppose 25 samples gave an average particle count of 150 per sample. Estimate the particle density, per cubic centimeter, and place a bound on the error of estimation (Points: 4) Suppose that in the water sampling question in #4, detecting the presence or absence of particles is easy Out counting particles is difficult. Among 750 such samples, 525 showed the particles (lime) to be present. Estimate the particle density and place a bound on the error of estimation. (Points: 4)

Answer 1

Answer:

Given Data

Consider a study variable are unknown .

ΧN N(μ, σ2) where μαnd σ

Here
(X, S
is jointly sufficient for
(8,0")

when  $X_{1},------,X_{n}$ be an rs of size n

(I) To test $H_{0}:\mu=\mu_{0}[\sigma\,\,is\,\,unknown]$ student's t- test

Here
X
is unbaised for $\mu$ . Under $H_{0}:\mu=\mu_{0}$ we expect that observed
X
should be very else to $\mu_{0}\,\,i.e\,\,\bar{X}\sim \mu_{0}$

Under $H_{0}:\mu=\mu_{0}\quad\quad$

$\left.\begin{matrix} \bar{X}\sim N(\mu_{0},\frac{\sigma^{2}}{n})\\ \frac{(n-1)S^{2}}{\sigma^{2}}\sim \chi^{2}n-1 \end{matrix}\right\}independent$

$\Rightarrow\frac{\frac{\bar{X}-\mu_{0}}{\sigma/\sqrt{n}}}{\sqrt{\frac{\frac{(n-1)S^{2}}{\sigma^{2}}}{(n-1)}}}\sim t_{n}-1$

$\Rightarrow T=\frac{\sqrt{n}(\bar{X}-\mu_{0})}{S}\sim t_{n}-1;$

the distribution is completely specified and is symmetric about '0'

Under $H_{0}:\mu=\mu_{0},$ we expect that observed

$T=\frac{\sqrt{n}(\bar{X}-\mu_{0})}{S}$

If observed

$\left | \frac{\sqrt{n}(\bar{X}-\mu_{0})}{S} \right |>e$, then we reject $H_{0}:\mu=\mu_{0}$ in favour of $H_{1}:\mu\neq\mu_{0}$ , where e is so choosen that $P_{H_{0}}\left [ \left | \frac{\sqrt{n}(\bar{X}-\mu_{0})}{S} \right |>e \right ]=\alpha$ , the choosen level of significance

$P[t_{n}>t_{\alpha;n}]=\alpha(upper\,\,\alpha\,\,point)\\Here\,\,P_{H_{0}}[|T|>e]=\alpha/2\\\Rightarrow e=t_{\alpha/2;n-1}$

Here , we reject $H_{0}:\mu=\mu_{0}$ vs  $H_{1}:\mu\neq\mu_{0}$ at level $\alpha$ , if observed

$\left | \frac{\sqrt{n}(\bar{X}-\mu_{0})}{S} \right |>t_{\alpha/2;n-1}$

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