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3 years ago

Which sample size will produce the widest 95% confidence interval, given a sample proportion of 0.5?

Mathematics

1 answer:

valentinak56 [21]3 years ago

3 0

Answer:

D. 50

Step-by-step explanation:

Width: 2z × sqrt(var/n)

Interval is wider when:

z is higher

n is smaller

In the options, least value of n is 50

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[tex]cos {}^{4} α+sin {}^{4} α= \frac{1}{4} (3+cos4α)<br>Prove:<br>

asambeis [7]

Given:

$\cos^4 \alpha+\sin^4\alpha=\dfrac{1}{4}(3+\cos 4 \alpha)$

To prove:

The given statement.

Proof:

We have,

$\cos^4 \alpha+\sin^4\alpha=\dfrac{1}{4}(3+\cos 4 \alpha)$

$LHS=\cos^4 \alpha+\sin^4\alpha$

$LHS=(\cos^2 \alpha)^2+(\sin^2 \alpha)^2$

$LHS=(\cos^2 \alpha+\sin^2\alpha)^2-2\sin ^2\alpha\cos^2 \alpha$ $[\because a^2+b^2=(a+b)^2-2ab]$

$LHS=(1)^2-2(1-\cos^2 \alpha)\cos^2 \alpha$ $[\because \cos^2 \alpha+\sin^2\alpha=1]$

$LHS=1-2\cos^2 \alpha+2\cos^4 \alpha$

Now,

$RHS=\dfrac{1}{4}(3+\cos 4 \alpha)$

$RHS=\dfrac{1}{4}[3+(2\cos^2 2\alpha-1)]$ $[\because \cos 2\theta=2\cos^2\theta -1]$

$RHS=\dfrac{1}{4}[2+2\cos^2 2\alpha]$

$RHS=\dfrac{1}{4}[2+2(2\cos^2 \alpha-1)^2]$ $[\because \cos 2\theta=2\cos^2\theta -1]$

$RHS=\dfrac{1}{4}[2+2(4\cos^4 \alpha-4\cos \alpha+1)]$ $[\because (a-b)^2=a^2-2ab+b^2]$

$RHS=\dfrac{1}{4}[2+8\cos^4 \alpha-8\cos \alpha+2]$

$RHS=\dfrac{1}{4}[4+8\cos^4 \alpha-8\cos \alpha]$

$RHS=1+2\cos^4 \alpha-2\cos \alpha$

$RHS=1-2\cos^2 \alpha+2\cos^4 \alpha$

$LHS=RHS$

Hence proved.

8 0

3 years ago

Solve each quadratic equation by factoring and using the zero product property.<br><br> n^2 - 49 = 0

Karo-lina-s [1.5K]

Answer:

x = 7 x = -7

Step-by-step explanation:

n^2 - 49 = 0

+49 +49

n^2 = 49

$\sqrt{n} = \sqrt{49} \\\\n = +/- 7\\$

7 0

3 years ago

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A clothing store is selling a pair of jeans for $17 and a sweatshirt for $8. Which inequality shows the number of sweatshirts Mo

vladimir2022 [97]

Answer:

1 pair of jeans and 3 sweatshirts

Step-by-step explanation:

41-17=24

24/8=3

3 0

3 years ago

A visitor makes a list of 11 sights he’d like to see. He is determined to visit at least 8 of these. How many possible choices d

Cerrena [4.2K]

Using the combination formula, it is found that he has 232 choices.

The order in which the sights are seen is not important, hence the <em>combination formula </em>is used to solve this question.

<h3>What is the combination formula?</h3>

$C_{n,x}$ is the number of different combinations of x objects from a set of n elements, given by:

$C_{n,x} = \frac{n!}{x!(n-x)!}$

In this problem, the choices are 8, 9, 10 or 11 sights from a set of 11, hence:

$T = C_{11,8} + C_{11,9} + C_{11,10} + C_{11,11} = \frac{11!}{3!8!} + \frac{11!}{2!9!} + \frac{11!}{10!1!} + \frac{11!}{0!11!} = 232$

He has 232 choices.

You can learn more about the combination formula at brainly.com/question/25821700

4 0

2 years ago

A paper popcorn cone measures 4 inches in diameter and 9 inches high. What is the volume of popcorn that can be contained within

Len [333]

Let me restate the given:

Diameter = 4 inches
height = 9 inches

The volume of a cone is formulated as V = (πr²h)/ 3

Since the formula uses a radius and the given is the diameter of a circle, then simply divide the diameter by 2 to get the radius.

r = 2 inches

Solve for volume of cone:

V = [ (3.14) (2)² (9) ] / 3
V = 37.7 in³

The answer is 37.70 in^3.

8 0

3 years ago

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