Answer: 106
Step-by-step explanation:
Let p be the true population proportion of cans in the warehouse that were contaminated by the fire.
The formula to find the required sample size when the prior populatio proportion is unknown:
[tex]n=0.25(\dfrac{z^*}{E})^2[/tex] , where z* = Critical z-value related to confidence interval and E = Margin of error.
As per given , we have
E= 0.08
Critical z-value for 90% confidence level = 1.645
Then, substitute all values in formula , we get
[tex]n=0.25(\dfrac{1.645}{0.08})^2\\\\=0.25(422.81640625)\\\\=105.704101563\approx106[/tex]
Hence, the statistician should sample 106 cans to estimate the proportion that were contaminated by the fire to within 0.08 with 90% confidence.
The p-value is a quantitative metric that is used to test a hypothesis against actual facts. so, the number of aluminum cans is "106".
The desired margin of error is [tex]ME = 0.08[/tex], and we pick [tex]Z_{0.05}= 1.645[/tex], a well-known number from the Z table, for [tex]90\%[/tex] confidence.
We choose the conservative estimate [tex]\hat{p} = 0.5[/tex] because we have no prior information on the fraction p.
We have the following:
[tex]\to \text{Margin of error} = Z_c \times \sqrt{\frac{\hat{P}(1 - \hat{p})}{n}}\\\\\to n = (\frac{Z_C}{\text{margin of error}})^2 \times \hat{p} (1 -\hat{p})\\\\[/tex]
[tex]=\left ( \frac{1.645}{0.08} \right )^{2}\times0.50(1-0.50)\\\\=\left (20.5625\right )^{2}\times0.50(1-0.50)\\\\=\left (20.5625\right )^{2}\times0.50(0.50)\\\\=105.40\approx 106\\\\[/tex]
For estimate p to within 0.08 with [tex]90\%[/tex] confidence, n =106 aluminum cans should be randomly picked.
Find out more about the p-value here:
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