There is a proposal before congress to build a so-called fourth-generation nuclear reactor at the idaho national laboratory. government officials want to estimate the true proportion of citizens in the state who support building the reactor. the most recent survey on idaho public opinion about building nuclear reactors showed 40% in favor. if the government officials want a margin of error of 4% and a confidence level of 90%, how many voters will they need to survey?

406

Step-by-step explanation:

Data provided in the question:

Margin of error, E = 4% = 0.04

Probability of people in favor, p = 40% = 0.4

Confidence level = 90%

Now,

For 90% confidence level, the z value = 1.645

let n be the required sample size

Thus,

n = p(1 - p)( z ÷ E)²

or

n = 0.4 × (1 - 0.4)( 1.645 ÷ 0.04 )²

or

n = 0.4 × 0.6 × 1691.265625

or

n = 405.90 ≈ 406

step-by-step explanation:

30% of $420 is 0.30($420) = $126 and $420 + $126 = $546.

or, 130% of $420 is 1.3($420) = $546.

notice that this is not the $600 original list price with which we started.

why doesn't this work?

the method does not work because 30% of $420 is not the same amount as the 30% of $600. it may seem obvious when stated this way, but it is a very common error made on problems of this type. we cannot just use the same percent and get back to the original amount because we are taking that percent of a different value.

to work problems of this type correctly, it is generally best to write an equation for the situation and then solve.

example: since (100% – 30%)(original price) = (sale price)

70% of the   (original price) is $420.

0.7x   = 420

equation image indicator

x = 600

the original cost was $600.

we may also solve this problem using the two-step method. working the problem in two steps requires us to use more algebra skills. we have to know that x = 1x (identity property of multiplicationtext annotation indicator) and that 1x – 0.3x = 0.7x (subtracting like terms). then we solve the problem as follows:

(original price) – 30%(original price) = sale price

x – 0.3x = 420

0.7x = $420

equation image indicator

x = 600

the original cost was $600.  

note that with both of the above methods we do obtain the correct original amount.

example: after a 5% pay raise, hermione is earning $22,680 per year. what was she earning before the pay raise?

105% of (original salary) = current salary

1.05x = 22,680

equation image indicator

x = 21,600

hermione earned $21,600 per year before the pay raise.

this is an example so maybe you will get how to at least do it!