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3 years ago

Sukie interviewed 125 employees at her company and discovered that 21 of them planned to take an extended vacation next year.

Mathematics

1 answer:

Mademuasel [1]3 years ago

4 0

Answer:

The standard error is 0.033.

Step-by-step explanation:

We are given that Sukie interviewed 125 employees at her company and discovered that 21 of them planned to take an extended vacation next year.

Let $\hat p$ = proportion of employees who planned to take an extended vacation next year

$\hat p$ = $\frac{X}{n}$ = $\frac{21}{125}$ = 0.168

n = number of employees at her company = 125

Now, the standard error is calculated by the following formula;

Standard error, S.E. = $\sqrt{\frac{\hat p(1-\hat p)}{n} }$

= $\sqrt{\frac{0.168(1-0.168)}{125} }$

= $\sqrt{\frac{0.168 \times 0.832}{125} }$ = 0.033

Hence, the standard error is 0.033.

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Answer:

θ = 0.896 and 2.246.

Step-by-step explanation:

We will use the identity

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1. Convert the equation to a more solvable form.

$\begin{array}{rcll}\sin\theta + 1 & = & \cos2\theta &\\\sin\theta + 1& = & 1-2\sin^{2}\theta & \text{Used the double angle formula}\\f(\theta) = 2\sin^{2}\theta + \sin\theta -2 & = & 0 & \text{Moved terms to left-hand side}\\2u^2 + u - 2 & = & 0 & \text{Substituted u for }\sin\theta\\\end{array}$

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$\begin{array}{rcl}u = \dfrac{-1 - 4.123}{4} & \qquad & u = \dfrac{-1 + 4.123}{4}\\\\u = -\dfrac{5.123}{4} & \qquad & u =\dfrac{3.123}{4}\\\\u = -1.281& \qquad & u =0.7808\\\end{array}$

3. Solve for θ

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u = sinθ = -1.281

sinθ cannot be less than -1. No solution.

(b) The positive root

If sin(x) = a, then

x = arcsin(x) + 2πn and

x = π - arcsin(a) + 2πn

(i) Case 1

u = sinθ = 0.7808

θ = arcsin(0.7808) + 2πn

θ = 0.8959 ± 2πn

If n = 0,

θ = 0.8959

(ii) Case 2

θ = π - 0.8959 + 2πn

If n = 0,

θ = π - 0.8959 = 2.246

The solutions are θ = 0.8959 and θ = 2.246

The diagram below shows the graph of f(θ) with its roots at 0.896 and 2.246.

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