Question

A new batching plant has been constructed and of the first 50 batches, three are found to not meet specification. a. Construct a 98% Score confidence interval for the population proportion of batches that do not meet specification for the new batching plant. b. Construct a 98% traditional confidence interval.c. Comment on the difference.

Answer 1

Answer:

The responses to these question can be defined as follows:

Step-by-step explanation:

$x=3 \\\\n=50\\\\\hat{p}=\frac{x}{n}=0.06\\\\\hat{q}=1-\hat{p}=0.94$

In point a:

$98\%$ confidence interval for population proportion (p):

$c=98\%=0.98\\\\\alpha=1-c=0.02\\\\\frac{\alpha}{2}=\frac{0.02}{2}=0.01\\\\z_{\frac{\alpha}{2}}=2.326$

For point b:

$98\% \ confidence\ interval =\hat{p} \pm z_{\frac{\alpha}{2}} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$

                                        $=0.06 \pm 2.326 \sqrt{\frac{0.06\times 0.94}{50}}\\\\ =0.06 \pm 2.326 (0.0336)\\\\=0.06 \pm 0.078\\\\98\% \ CI =(-0.018, 0.138)$

For point c:

$The\ difference\ between \ 98\% \ CI\ is \ -0.018\ to\ 0.138$