a survey conducted by the u.s department of labor found the 48 out of 500 heads of house holds were unemployed compute a 99% confidence interval for the proportion of unemployed heads of households what is the upper and lower limits round the answers to three decimals
To compute the 99% confidence interval for the proportion of unemployed heads of households, you can use the formula for confidence intervals for proportions:
Confidence Interval = Sample Proportion ± Margin of Error
Here's how to calculate each component:
1. Sample Proportion (p̂):
The sample proportion is the ratio of the number of unemployed heads of households to the total number of heads of households surveyed. In this case, 48 out of 500 heads of households were unemployed.
Sample Proportion (p̂) = Number of Unemployed / Total Sample Size
p̂ = 48 / 500
p̂ = 0.096
2. Margin of Error (E):
The margin of error represents the uncertainty associated with the calculated sample proportion. It can be determined using the following formula:
Margin of Error (E) = Z * sqrt(p̂ * (1 - p̂) / n)
Z represents the z-score corresponding to the desired confidence level. For a 99% confidence level, the z-score is approximately 2.576.
n is the total sample size, which is 500 in this case.
Substituting the values into the formula:
E = 2.576 * sqrt(0.096 * (1 - 0.096) / 500)
E ≈ 0.035
3. Confidence Interval:
Finally, you can compute the confidence interval by subtracting and adding the margin of error from the sample proportion:
Lower Limit = p̂ - E
Upper Limit = p̂ + E
Lower Limit = 0.096 - 0.035
Lower Limit ≈ 0.061
Upper Limit = 0.096 + 0.035
Upper Limit ≈ 0.131
Therefore, the 99% confidence interval for the proportion of unemployed heads of households is approximately 0.061 to 0.131.
To compute a confidence interval for the proportion of unemployed heads of households, we can use the formula:
CI = p̂ ± Z * sqrt(p̂(1-p̂) / n)
Where:
CI = Confidence Interval
p̂ = Proportion of heads of households who were unemployed (48/500 = 0.096)
Z = Z-score for the desired confidence level (for 99% confidence level, Z ≈ 2.576)
n = Total number of heads of households (500)
Let's calculate the confidence interval:
CI = 0.096 ± 2.576 * sqrt(0.096 * (1 - 0.096) / 500)
CI = 0.096 ± 2.576 * sqrt(0.096 * 0.904 / 500)
CI = 0.096 ± 2.576 * sqrt(0.086784 / 500)
CI = 0.096 ± 2.576 * sqrt(0.000173568)
CI = 0.096 ± 2.576 * 0.013158
CI = 0.096 ± 0.033949
Now, we can round the upper and lower limits to three decimals:
Upper Limit = 0.096 + 0.033 = 0.129
Lower Limit = 0.096 - 0.034 = 0.062
Therefore, the 99% confidence interval for the proportion of unemployed heads of households is approximately [0.062, 0.129].