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Math  /  Data & Statistics

QuestionAn epidemiologist plans to conduct a survey to estimate the percentage of women who give birth. How many women must be surveyed in order to be 95%95 \% confident that the estimated percentage is in error by no more than one percentage point? Complete parts (a) through (c) below. a. Assume that nothing is known about the percentage to be estimated. n=□\mathrm{n}=\square (Round up to the nearest integer.) b. Assume that a prior study conducted by an organization showed that 81%81 \% of women give birth. n=n= □\square (Round up to the nearest integer.) c. What is wrong with surveying randomly selected adult women? A. Randomly selecting adult women would result in an overestimate, because some women will give birth to their first child after the survey was conducted. It will be important to survey women who have completed the time during which they can give birth. B. Randomly selecting adult women would result in an overestimate, because some women will give birth to their first child after the survey was conducted. It will be important to survey women who have already given birth. C. Randomly selecting adult women would result in an underestimate, because some women will give birth to their first child after the survey was conducted. It will be important to survey women who have already given birth. D. Randomly selecting adult women would result in an underestimate, because some women will give birth to their first child after the survey was conducted. It will be important to survey women who have completed the time during which they can give birth.

Studdy Solution

STEP 1

1. The confidence level is 95% 95\% , corresponding to a z z -score of approximately 1.96 1.96 .
2. The margin of error is 1% 1\% or 0.01 0.01 .
3. For part (a), no prior estimate of the proportion is available.
4. For part (b), a prior estimate of the proportion is 81% 81\% or 0.81 0.81 .

STEP 2

1. Calculate the sample size when nothing is known about the proportion.
2. Calculate the sample size using a prior estimate of the proportion.
3. Evaluate the implications of randomly selecting adult women.

STEP 3

For part (a), use the formula for sample size when no prior estimate is available:
n=(zE)2×14 n = \left( \frac{z}{E} \right)^2 \times \frac{1}{4}
Where z=1.96 z = 1.96 and E=0.01 E = 0.01 .

STEP 4

Calculate the sample size:
n=(1.960.01)2×14 n = \left( \frac{1.96}{0.01} \right)^2 \times \frac{1}{4}
n=(196)2×14 n = \left( 196 \right)^2 \times \frac{1}{4}
n=38416×14 n = 38416 \times \frac{1}{4}
n=9604 n = 9604
Round up to the nearest integer:
n=9604 n = 9604

STEP 5

For part (b), use the formula for sample size with a prior estimate:
n=(zE)2×p(1−p) n = \left( \frac{z}{E} \right)^2 \times p(1-p)
Where p=0.81 p = 0.81 .

STEP 6

Calculate the sample size:
n=(1.960.01)2×0.81×(1−0.81) n = \left( \frac{1.96}{0.01} \right)^2 \times 0.81 \times (1 - 0.81)
n=38416×0.81×0.19 n = 38416 \times 0.81 \times 0.19
n=38416×0.1539 n = 38416 \times 0.1539
n=5912.5824 n = 5912.5824
Round up to the nearest integer:
n=5913 n = 5913

STEP 7

Evaluate the implications of randomly selecting adult women.

STEP 8

Option D is correct. Randomly selecting adult women would result in an underestimate, because some women will give birth to their first child after the survey was conducted. It will be important to survey women who have completed the time during which they can give birth.
The solutions are: a. n=9604 n = 9604 b. n=5913 n = 5913 c. Option D

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