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Mathematics

1 answer:

Scrat [10]3 years ago

3 0

Answer:

Correct answer: ± √(1+x)/2 = cos Θ/2

Step-by-step explanation:

We know it is:

cos2Θ = cos²Θ - sin²Θ, according to this:

cosΘ = cos²Θ/2 - sin²Θ/2 => cosΘ = cos²Θ/2 - (1 - cos²Θ/2) =>

cosΘ = cos²Θ/2 - 1 + cos²Θ/2 => 1 + cosΘ = 2 cos²Θ/2 =>

cos²Θ/2 = (1 + cosΘ)/2 => cos Θ/2 = ±√(1 + cosΘ)/2 =>

±√(1 + cosΘ)/2 = ± √(1+x)/2 = cos Θ/2

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In a study of government financial aid for college students, it becomes necessary to estimate the percentage of full-time coll

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Answer:

(a) The sample size required is 2401.

(b) The sample size required is 2377.

Step-by-step explanation:

The confidence interval for population proportion p is:

$CI=\hat p\pm z_{\alpha/2}\sqrt{\frac{\hatp(1-\hat p)}{n}}$

The margin of error in this interval is:

$MOE=z_{\alpha/2}\sqrt{\frac{\hatp(1-\hat p)}{n}}$

The information provided is:

MOE = 0.02

$z_{\alpha/2}=z_{0.05/2}=z_{0.025}=1.96$

(a)

Assume that the proportion value is 0.50.

Compute the value of n as follows:

$MOE=z_{\alpha/2}\sqrt{\frac{\hat p(1-\hat p)}{n}}\\0.02=1.96\times \sqrt{\frac{0.50(1-0.50)}{n}}\\n=\frac{1.96^{2}\times0.50(1-0.50)}{0.02^{2}}\\=2401$

Thus, the sample size required is 2401.

(b)

Given that the proportion value is 0.55.

Compute the value of n as follows:

$MOE=z_{\alpha/2}\sqrt{\frac{\hat p(1-\hat p)}{n}}\\0.02=1.96\times \sqrt{\frac{0.55(1-0.55)}{n}}\\n=\frac{1.96^{2}\times0.55(1-0.55)}{0.02^{2}}\\=2376.99\\\approx2377$