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7.40

Expert-verifiedFound in: Page 300

Book edition
9th

Author(s)
Weiss, Neil

Pages
590 pages

ISBN
9780321989505

Refer to Exercise $7.10$ on page $295$.

a. Use your answers from Exercise $7.10\left(b\right)$ to determine the mean, ${\mu}_{i}$, of the variable $\hat{x}$ for each of the possible sample sizes.

b. For each of the possible sample sizes, determine the mean, ${\mu}_{i+}$ of the variable $\hat{x}$, using only your answer from Exercise $7.10\left(a\right)$

- The mean ${\mu}_{\overline{x}}$ of the variable $\overline{x}$ will be $5$.
- The population mean of the given data is $5$

Given in the question that

Population data: $2,3,5,5,7,8$.

The population data for the variable $x$is $2,3,5,5,7,8$

Here determine the role="math" localid="1651232434578" ${\mu}_{\overline{x}}$ of the variable $\overline{x}$ for each sample in the question

The possible sample and sample means for a sample of size $n=1$ is given in the table

$\begin{array}{|ll|}\hline \text{Sample}& \overline{x}\\ 2& 2\\ 3& 3\\ 5& 5\\ 5& 5\\ 7& 7\\ 8& 8\\ \hline\end{array}$

Therefore the mean ${\mu}_{\overline{x}}$ of the variable $\overline{x}$will be,

role="math" localid="1651232793762" ${\mu}_{\overline{x}}=\frac{2+3+5+5+7+8}{6}\phantom{\rule{0ex}{0ex}}=\frac{30}{6}\phantom{\rule{0ex}{0ex}}={5}^{6}$

The mean ${\mu}_{\overline{x}}$ of the variable $\overline{x}$ will be $5$

Now we need to calculate the possible sample and sample means for sample of size $n=2$

The table will be

$\begin{array}{|llll|}\hline \text{Sampie}& \overline{x}& \text{Sample}& \overline{x}\\ 2.3& 2.5& 3.7& 5\\ 2.5& 3.5& 3.8& 5.5\\ 2.5& 3.5& 5.5& 5\\ 2,7& 4.5& 5,7& 6\\ 2,8& 5& 5,8& 6.6\\ 3,5& 4& 5,7& 6\\ 3.5& 4& 5.8& 6.5\\ & & 7.8& 7.5\\ \hline\end{array}$

The mean ${\mu}_{\overline{x}}$ for the variable $\overline{x}$ will be

role="math" localid="1651234078644" ${\mu}_{x}=\frac{\left\{\begin{array}{l}3.3+3.3+4+4.3+4+4.7+5+4.7+5+5.7+4.3+5+5.3+5+\\ 5.3+6+5.7+6+6.7+6.7\end{array}\right\}}{20}\phantom{\rule{0ex}{0ex}}=\frac{100}{20}\phantom{\rule{0ex}{0ex}}=5$

The mean ${\mu}_{\overline{x}}$ of the variable $\overline{x}$ when sample size $n=2$ is $5$

Given in the question that

Population data: $2,3,5,5,7,8$.

Here we need to find the population mean

The calculation for the population mean will be,

$\mu =\frac{\sum {x}_{i}}{N}\phantom{\rule{0ex}{0ex}}=\frac{2+3+5+5+7+8}{6}\phantom{\rule{0ex}{0ex}}=5$

So, The population mean will be $5$.

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