A 95% confidence interval for a difference in proportions p1-p2 if the samples have n1=70 with p^1=0.64 and n2=90 with p^2=0.58, and the standard error is SE=0.08.
To calculate the 95% confidence interval for the difference in proportions (p₁ - p₂), you can use the formula:
CI = (p₁ - p₂) ± Z * SE
Where:
p₁ = proportion in sample 1
p₂ = proportion in sample 2
SE = standard error
Z = Z-score for the desired confidence level (for 95% confidence, Z ≈ 1.96)
Given the values n₁ = 70, p̂₁ = 0.64, n₂ = 90, p̂₂ = 0.58, and SE = 0.08, we can substitute these values into the formula:
CI = (0.64 - 0.58) ± 1.96 * 0.08
Simplifying the equation:
CI = 0.06 ± 1.96 * 0.08
Calculating the values:
CI = 0.06 ± 0.1568
Which gives us two intervals:
Lower bound: 0.06 - 0.1568 = -0.0968
Upper bound: 0.06 + 0.1568 = 0.2768
Therefore, the 95% confidence interval for the difference in proportions (p₁ - p₂) is approximately -0.0968 to 0.2768.
To compute the 95% confidence interval for a difference in proportions (p1 - p2), you can use the formula:
CI = (p1 - p2) ± (Z * SE)
where:
- p1 and p2 are the sample proportions of the two groups
- SE is the standard error of the difference in proportions
- Z is the z-score corresponding to the desired confidence level (in this case, 95%)
Given that the standard error is SE = 0.08, and the samples have n1 = 70 with p^1 = 0.64 and n2 = 90 with p^2 = 0.58, we can calculate the confidence interval.
First, let's compute the z-score for a 95% confidence interval. The z-score corresponds to the area under the standard normal distribution curve. For a 95% confidence level, you need to find the z-score that leaves 2.5% in each tail of the distribution. This can be done using a z-score table or a statistical software. The z-score for a 95% confidence interval is approximately 1.96.
Now we can substitute the values into the formula and calculate the confidence interval:
CI = (p1 - p2) ± (Z * SE)
= (0.64 - 0.58) ± (1.96 * 0.08)
= 0.06 ± 0.1568
Calculating further:
Lower bound = 0.06 - 0.1568 = -0.0968
Upper bound = 0.06 + 0.1568 = 0.2768
Therefore, the 95% confidence interval for the difference in proportions (p1 - p2) is (-0.0968, 0.2768).