Confidence Intervals; In a survey of 1004 individuals, 442 felt that Randolph spent too much time away. Find a 95% confidence interval for the true population proportion
Use a proportional confidence interval formula:
CI95 = p + or - (1.96)(√pq/n)
Note: + or - 1.96 represents 95% confidence interval.
For p in your problem: 442/1004
For q: 1 - p
n = 1004
Be sure to convert all fractions to decimals. It's easier to work with decimals in the formula.
I let you take it from here to calculate the interval.
To find the 95% confidence interval for the true population proportion, we can use the formula:
CI = p̂ ± Z * √((p̂(1 - p̂)) / n)
Where:
- p̂ is the sample proportion (442/1004)
- Z is the Z-score corresponding to the desired confidence level (95% confidence level corresponds to a Z-score of 1.96)
- n is the sample size (1004)
Let's calculate it step by step:
1. Find the sample proportion:
p̂ = (number of individuals who felt Randolph spent too much time away) / (total sample size)
= 442 / 1004
≈ 0.4392
2. Determine the Z-score for a 95% confidence level:
Since we want a 95% confidence level, the corresponding Z-score is 1.96. This value represents how many standard deviations away from the mean we need to be in order to capture 95% of the data.
3. Calculate the standard error:
The standard error gives us an idea of the amount of variability we can expect in our sample proportion.
Standard error (SE) = √((p̂(1 - p̂)) / n)
= √((0.4392(1 - 0.4392)) / 1004)
≈ 0.0166
4. Calculate the margin of error (ME):
The margin of error tells us the maximum likely difference between the true population proportion and our sample proportion.
Margin of error (ME) = Z * SE
= 1.96 * 0.0166
≈ 0.0325
5. Calculate the confidence interval (CI):
The confidence interval is calculated by taking the sample proportion and adding or subtracting the margin of error.
CI = p̂ ± ME
= 0.4392 ± 0.0325
≈ (0.4067, 0.4717)
Therefore, the 95% confidence interval for the true population proportion is approximately (0.4067, 0.4717). This means that we can be 95% confident that the true proportion of individuals who feel that Randolph spends too much time away falls within this range.