Based on the question, here are the given information:
n = 40 mice as sample
x ≤ 30 seconds
proportion (p) = 42% or 0.42 in decimal form
Find: 90% limit or confidence interval
Solution:
The formula in getting the confidence interval in proportion is:
[tex]CI=p\pm Z(\sqrt[]{\frac{p(1-p)}{n}}[/tex]
Recall that for 90% confidence interval, the z-value is 1.645. Since we already have the values of "p" and "n" given in the problem, let's plug these values into the formula above.
[tex]CI=0.42\pm1.645\sqrt[]{\frac{0.42(1-0.42)}{40}}[/tex]
Then, solve.
[tex]\begin{gathered} CI=0.42\pm1.645\sqrt[]{\frac{0.42\times0.58}{40}} \\ CI=0.42\pm1.645\sqrt[]{\frac{0.2436}{40}} \\ CI=0.42\pm1.645(0.078038) \\ CI=0.42\pm0.12837 \end{gathered}[/tex]
Separate the plus and minus signs.
[tex]\begin{gathered} CI=0.42+0.12837=0.54837\Rightarrow54.8percent \\ CI=0.42-0.12837=0.29163\Rightarrow29.2percent \end{gathered}[/tex]
Hence, the 90% limit of the proportion is between 29.2% to 54.8%. Based on the options, the answer is 29.1% < p < 54.9% (third option).
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