A researcher wishes to estimate the proportion of adults who have high-spped internet access. What size sample should be obtained if she wishes the estimate to be within .02 with 90% if
A) She uses a previous estimate of 0.48?
B) She does not use any prior estimates?
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A researcher wishes to estimate the proportion of adults who have high-spped internet access. What size sample should be obtained if she wishes the estimate to be within 1% of true proportion
How do you know what the z-value is? I've never used a z-table.
To determine the sample size needed to estimate the proportion of adults with high-speed internet access, we can use a formula called the sample size formula for estimating a proportion. This formula takes into account the desired level of precision, the desired level of confidence, and any prior estimates of the proportion.
A) Using a Previous Estimate:
1. Start by identifying the known values from the problem:
- Desired level of precision: 0.02
- Desired level of confidence: 90% (which corresponds to a z-value of 1.645)
- Previous estimate: 0.48
2. Calculate the estimated standard deviation (sd) using the previous estimate:
sd = sqrt[(p_hat * q_hat) / n]
where p_hat is the previous estimate, q_hat is 1 - p_hat, and n is the sample size.
3. To be conservative, we can use the worst-case scenario by assuming the previous estimate yields the largest standard deviation, which occurs when p_hat = 0.5:
sd = sqrt[(0.5 * 0.5) / n] = sqrt(0.25 / n) = 0.5/sqrt(n)
4. Determine the sample size using the formula for estimating a proportion:
n = (z^2 * p_hat * q_hat) / E^2
where z is the z-value corresponding to the desired level of confidence, p_hat is the previous estimate, q_hat is 1 - p_hat, and E is the desired level of precision.
Substituting known values:
n = [(1.645)^2 * 0.48 * 0.52] / (0.02)^2
n = [2.706 * 0.2496] / 0.0004
n ≈ 1770
Therefore, a sample size of at least 1770 should be obtained if the researcher wishes the estimate to be within 0.02 with 90% confidence, using the previous estimate of 0.48.
B) Without Using any Prior Estimates:
If the researcher does not use any prior estimates, we can assume the most conservative scenario where p_hat = q_hat = 0.5. This yields the largest possible sample size. Using the same formula as above, n = (z^2 * p_hat * q_hat) / E^2:
n = [(1.645)^2 * 0.5 * 0.5] / (0.02)^2
n = [2.706 * 0.25] / 0.0004
n = 1691
Therefore, a sample size of at least 1691 should be obtained if the researcher does not use any prior estimates and wishes the estimate to be within 0.02 with 90% confidence.
It's worth noting that these calculations assume a simple random sampling method and may need adjustment if more complex sampling techniques are used.
Formula to find sample size:
n = [(z-value)^2 * p * q]/E^2
... where n = sample size, z-value is found using a z-table for 90% confidence (which is 1.645), p = .48 (when no value is stated in the problem p = .50), q = 1 - p, ^2 means squared, * means to multiply, and E = .02.
Plug values into the formula and calculate n.
I hope this will help get you started.