1) Construct a confidence interval of the population proportion at the given level of confidence.
x=60,n=300, 99% confidence.
The 99% confidence interval is (_,_)
2) A researcher wishes to estimate the proportion of adults who have high-speed internet access. What size sample should be obtained if she wishes the estimate to be within 0.04 with 99% confidence if
a) she uses a previous estimate of 0.52?
b) she does not use any prior estimates?
1. 99% = mean ± 2.575 SEm
SEm = SD/√n
What is the value of the standard deviation (SD)?
1) To construct a confidence interval for the population proportion, we can use the formula:
Confidence Interval for Population Proportion = Sample Proportion ± Margin of Error
Given:
Sample Proportion (p̂) = x/n = 60/300 = 0.2 (proportion of successes in the sample)
Level of Confidence (C) = 99% (we want a 99% confidence interval)
To find the margin of error, we need to use the formula:
Margin of Error (E) = Critical Value * Standard Error
Since the sample size is large (n > 30) and we're dealing with proportions, we can use the z-distribution and the formula for the standard error of proportion:
Standard Error (SE) = √[(p̂ * (1-p̂))/n]
First, we need to find the critical value corresponding to a 99% confidence level. Looking up in the Standard Normal Distribution Table for a 99% confidence level (which corresponds to 1 - (1-99)/2 = 0.995), the critical value is approximately 2.576.
Now, we can substitute the values into the formula:
Standard Error (SE) = √[(0.2 * (1-0.2))/300] ≈ 0.0245
Margin of Error (E) = 2.576 * 0.0245 ≈ 0.0630
Confidence Interval = Sample Proportion ± Margin of Error = 0.2 ± 0.0630
Therefore, the 99% confidence interval is approximately (0.137, 0.263).
2) a) If the researcher uses a previous estimate of 0.52, to determine the sample size needed for a 99% confidence level and a margin of error of 0.04, we can use the following formula:
Sample Size (n) = (Z² * p̂ * (1-p̂))/(E²)
Given:
Z = Critical value corresponding to the desired confidence level (99% confidence corresponds to a Z-value around 2.576)
p̂ = Previous estimate = 0.52
E = Margin of error = 0.04
Substituting the values into the formula:
n = (2.576² * 0.52 * (1 - 0.52))/(0.04²) ≈ 668.24
Rounding up to the nearest whole number, the researcher should obtain a sample size of approximately 669.
b) If the researcher does not use any prior estimates, we can use a conservative estimate of 0.5 for p̂ (since p̂ = 0.5 yields the maximum sample size required). Using the same formula as before:
n = (2.576² * 0.5 * (1 - 0.5))/(0.04²) ≈ 676.92
Rounding up to the nearest whole number, the researcher should obtain a sample size of approximately 677.
1) To construct a confidence interval for a population proportion, we can use the formula:
CI = p̂ ± Z * √(p̂(1-p̂)/n)
Where:
- CI is the confidence interval
- p̂ is the sample proportion (x/n)
- Z is the critical value for the desired confidence level (in this case, 2.576 for a 99% confidence level)
- n is the sample size
Given x = 60, n = 300, and a 99% confidence level, we can calculate the confidence interval:
p̂ = (x / n) = (60 / 300) = 0.2
CI = 0.2 ± 2.576 * √((0.2 * (1-0.2)) / 300)
Calculating the values inside the square root:
= 0.2 ± 2.576 * √((0.04) / 300)
= 0.2 ± 2.576 * 0.02081
Calculating the result:
= 0.2 ± 0.05366
The 99% confidence interval is (0.14634, 0.25366).
2) a) When using a previous estimate of 0.52, we can use the formula for sample size:
n = (Z^2 * p̂ * (1-p̂)) / E^2
Where:
- n is the required sample size
- Z is the critical value for the desired confidence level (2.576 for a 99% confidence level)
- p̂ is the estimated proportion (0.52 in this case)
- E is the desired margin of error (0.04)
Plugging in the given values:
n = (2.576^2 * 0.52 * (1-0.52)) / 0.04^2
Calculating the numerator:
= (6.635776 * 0.52 * 0.48) / 0.0016
= 2.03488832 / 0.0016
Calculating the result:
n = 1271.8052
Therefore, a sample size of approximately 1272 should be obtained.
b) When no prior estimate is used, we can use the formula for sample size:
n = (Z^2) / E^2
Where:
- n is the required sample size
- Z is the critical value for the desired confidence level (2.576 for a 99% confidence level)
- E is the desired margin of error (0.04)
Plugging in the given values:
n = (2.576^2) / 0.04^2
= 6.635776 / 0.0016
Calculating the result:
n = 4147.36
Therefore, a sample size of approximately 4148 should be obtained.