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1. Find the sample size needed to estimate the proportion of voters favoring a candidate if we want our estimate to be within 3% with 95% confidence. A)2430 B)1068 C)89%

2. A political action group wishes to learn the government approval rating on the environment. From a past study they know that they will have to poll 270 people for their desired level of confidence. If they want to keep the same level of confidence but divide the margin of error in third, how many people will they have to poll? A)2430 B)1068 C)89%

3. A newspaper article indicates that an estimate of the unemployment rate involves a sample of 47,000 people. If the reported rate must be within 0.2% and the rate is known to be 8%, find the corresponding level of confidence. A)2430 B)1068 C)89%

Sagot :

Using the z-distribution, as we are working with proportions, it is found that:

1. 1068 people have to be sampled.

2. 2430 people have to be sampled.

3. A level of confidence of 89% is used.

What is a confidence interval of proportions?

A confidence interval of proportions is given by:

[tex]\pi \pm z\sqrt{\frac{\pi(1-\pi)}{n}}[/tex]

In which:

  • [tex]\pi[/tex] is the sample proportion.
  • z is the critical value.
  • n is the sample size.

The margin of error is:

[tex]M = z\sqrt{\frac{\pi(1-\pi)}{n}}[/tex]

Item a:

There is no estimate for the proportion, hence [tex]\pi = 0.5[/tex] is used. Considering a margin of error of 3% with 95% confidence, we have that [tex]M = 0.03, z = 1.96[/tex], hence:

[tex]M = z\sqrt{\frac{\pi(1-\pi)}{n}}[/tex]

[tex]0.03 = 1.96\sqrt{\frac{0.5(0.5)}{n}}[/tex]

[tex]0.03\sqrt{n} = 1.96(0.5)[/tex]

[tex]\sqrt{n} = \frac{1.96(0.5)}{0.03}[/tex]

[tex](\sqrt{n})^2 = \left(\frac{1.96(0.5)}{0.03}\right)^2[/tex]

[tex]n = 1067.1[/tex]

Rounding up, 1068 people have to be sampled.

Item 2:

The margin of error is inverse proportional to the square root of the sample size, hence to reduce the margin of error in third, the sample size has to be multiplied by 9, that is:

9 x 270 = 2430 people have to be sampled.

Item 3:

We have that:

[tex]n = 47000, M = 0.002, \pi = 0.08[/tex].

Hence:

[tex]M = z\sqrt{\frac{\pi(1-\pi)}{n}}[/tex]

[tex]0.002 = z\sqrt{\frac{0.08(0.92)}{47000}}[/tex]

[tex]0.00125z = 0.002[/tex]

[tex]z = 1.6[/tex]

Which is the critical value corresponding to a confidence level of 89%.

More can be learned about the z-distribution at https://brainly.com/question/25890103

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