1. A report states that 46% of home owners have a vegetable garden. How large a
ID: 3219056 • Letter: 1
Question
1. A report states that 46% of home owners have a vegetable garden. How large a sample is needed to estimate the true proportion of home owners who have vegetable gardens to within 4 percentage points with 98% confidence?
248
422
653
843
2. A random sample of 70 voters found that 40% were going to vote for a certain candidate. Find the 90% limit for the population proportion of voters who will vote for that candidate.
30.3% < p < 49.7%
31.4% < p < 48.6%
32.5% < p < 47.5%
35.2% < p < 44.8%
3. The Academy of Orthopedic Surgeons states that 80% of women wear shoes that are too small for their feet. A researcher wants to be 98% confident that this proportion is within 3 percentage points of the true proportion. How large a sample is necessary?
966
683
1183
484
248
422
653
843
2. A random sample of 70 voters found that 40% were going to vote for a certain candidate. Find the 90% limit for the population proportion of voters who will vote for that candidate.
30.3% < p < 49.7%
31.4% < p < 48.6%
32.5% < p < 47.5%
35.2% < p < 44.8%
3. The Academy of Orthopedic Surgeons states that 80% of women wear shoes that are too small for their feet. A researcher wants to be 98% confident that this proportion is within 3 percentage points of the true proportion. How large a sample is necessary?
966
683
1183
484
Explanation / Answer
1.
z = 2.33 for 98% CI, E = 0.04, p = 0.46, q = 0.54
From the sample size calculation formula
n = [z/E]^2*pq
n = [2.33/0.04]^2*0.46*0.54
n = 58.25 ^ 2 * 0.2484
3393.06 X 0.2484 = 842.83
n = 843 (round up)
2.
p = 0.40, q = 0.60, n = 70, Z = 1.645 for 90% CI
Putting in confidence interval formula
Lower limit = 0.40 - 1.645* (pq/n) = 0.40 - 1.645 * 0.24/70 = 0.40- 1.645* 0.003428571 = 0.40 - 1.645*0.058554 = 0.40 - 0.09632133 = 0.303
Upper limit = 0.40 + 1.645* (pq/n) = 0.40 + 1.645 * 0.24/70 = 0.40 + 1.645* 0.003428571 = 0.40 + 1.645*0.058554 = 0.40 + 0.09632133 = 0.497
Therefore, 90 % CI = 30.3% to 49.7%
3.
z = 2.33 for 98% CI, E = 0.03, p= 0.80, q= 0.20
Sample size calculation formula
n = [z/E]^2*pq
n = [2.33/0.03]^2*0.80*0.20
n = 77.66 ^ 2 * 0.16
6032.11 * 0.16 = 965.4
n = 966 (round up)
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