What is the Standard Deviation (SD)? Do you mean "x=4821" or ∑x = 4821 to calculate the mean?
99% = mean ± 2.575 SEm
SEm = SD/√n
99% = mean ± 2.575 SEm
SEm = SD/√n
Step 1: Identify the sample size (n) and the number of successes (x) in the sample.
Given in this case:
n = 5200 (sample size)
x = 4821 (number of successes)
Step 2: Determine the confidence level.
In this case, the confidence level is 99%.
A confidence level of 99% means that we are 99% confident that the true population proportion falls within the calculated confidence interval.
Step 3: Calculate the sample proportion (p-hat).
The sample proportion (p-hat) is calculated by dividing the number of successes (x) by the sample size (n):
p-hat = x / n
In this case:
p-hat = 4821 / 5200 = 0.9279
Step 4: Calculate the margin of error (E).
The margin of error (E) is determined by multiplying the critical value (z) with the standard error (SE).
The critical value (z) is related to the confidence level chosen. For a 99% confidence level, the critical value is approximately 2.576.
The standard error (SE) is calculated as follows:
SE = sqrt((p-hat * (1 - p-hat)) / n)
In this case:
SE = sqrt((0.9279 * (1 - 0.9279)) / 5200) = 0.00528
Now we can calculate the margin of error:
E = z * SE
E = 2.576 * 0.00528 = 0.0136
Step 5: Calculate the lower and upper limits of the confidence interval.
To determine the lower limit, subtract the margin of error from the sample proportion:
Lower limit = p-hat - E
Lower limit = 0.9279 - 0.0136 = 0.9143
To determine the upper limit, add the margin of error to the sample proportion:
Upper limit = p-hat + E
Upper limit = 0.9279 + 0.0136 = 0.9415
Step 6: Determine the confidence interval.
The confidence interval for the population proportion (p) is found by combining the lower and upper limits:
Confidence interval = (Lower limit, Upper limit)
Confidence interval = (0.9143, 0.9415)
Therefore, the confidence interval estimate of the population proportion (p) is approximately 0.9143 to 0.9415 at a 99% confidence level.