It costs you \(10 to draw a sample of size n = 1 and measure the attribute of interest. You have a budget of \)1,500.

a. Do you have sufficient funds to estimate the population mean for the attribute of interest with a 95% confidence interval 5 units in width? Assume$\mathbf{\sigma }\mathbf{=}\mathbf{14}$.

b. If you used a 90% confidence level, would your answer to part a change? Explain.

Given it costs $ 10 to draw a sample size$n=1$ and measure the attribute of interest.

a.

Here assume that$\sigma =14$.

Known that the width of 5 units.

Then, the marginal error (ME) is,

$\begin{array}{c}ME=\frac{5}{2}\\ =2.5\end{array}$

For a 95% confidence interval, the level of significance is $\alpha =0.05$.

Then,

$\begin{array}{c}\frac{\alpha }{2}=\frac{0.05}{2}\\ \frac{\alpha }{2}=0.025\end{array}$

Using the normal distribution tables,

$z_{0.025}=1.960$

The sample size (n) obtained by,

$\begin{array}{c}n={\left(\frac{z_{\frac{\alpha }{2}}\sigma }{ME}\right)}^2\\ ={\left(\frac{1.960\times 14}{2.5}\right)}^2\\ ={\left(\frac{27.44}{2.5}\right)}^2\\ ={\left(10.976\right)}^2\\ =120.472576\end{array}$

Hence, there is 121 samples which cost$121\left(\$10\right)=\$1210$,

Yes, $1210 is sufficient funds to estimate the population mean for the attributes interest with 95% confidence interval of 5 in width.

b.

For a 95% confidence interval, the level of significance is $\alpha =0.10$.

Then,

$\begin{array}{c}\frac{\alpha }{2}=\frac{0.10}{2}\\ \frac{\alpha }{2}=0.05\end{array}$

Using the normal distribution tables,

$z_{0.05}=1.645$

The sample size (n) obtained by,

$\begin{array}{c}n={\left(\frac{z_{\frac{\alpha }{2}}\sigma }{ME}\right)}^2\\ ={\left(\frac{1.645\times 14}{2.5}\right)}^2\\ ={\left(\frac{23.03}{2.5}\right)}^2\\ ={\left(9.212\right)}^2\\ =84.860944\end{array}$

$n\approx 85$

Hence, there is 85 samples which cost,$85\left(\$10\right)=\$850$

Yes, still $850 is sufficient funds to estimate the population mean for the attributes interest with 90% confidence level 5 in width.

So, the answer to part (a) is not changed.