I am working on a question/ problem that goes as follows:
I thought that I had understood the concepts behind the question, however, when plugging my answers within the system, I was notified that all of them were wrong and I cannot figure out why.
I was hoping that someone could help me pinpoint my mistakes and guide me towards the correct answers.
Here is my work:
Part (a)
Margin of Error, E $$= \frac{23.0132-22.6868}{2} =0.1632$$
$$E = z \times \frac{\sigma}{\sqrt{n}}$$ $$0.1632 = z \times \frac{0.40}{\sqrt{26}}$$ $$ z = \frac{0.1632 \times \sqrt{26}}{0.40} = 2.0804$$
From the z table, 96.2511% is associated with the Confidence Interval provided
Part (b)
Since the Z score for 90% confidence interval is 1.645,
$$P(-1.645 < Z < 1.645) = 0.90$$
$$E = z \times \frac{\sigma}{\sqrt{n}} = 1.645 \times \frac{0.40}{\sqrt{26}} = 0.12904$$
Confidence Interval for 90% is $$ \mu \pm E = 22.85 \pm 0.12904 =[22.720, 22.97904]$$
Part (c)
Since the Z score for 90% confidence interval is 2.576,
$$P(-2.576 < Z < 2.576) = 0.99$$
And, with the margin of error being the same as in part (a),
$$E = 0.1632$$ $$0.1632 = 2.576 \times \frac{0.40}{\sqrt{n}}$$ $$n = \left( \frac{2.576 \times 0.40}{0.1632}\right)^2 = 39.86 \sim 40$$