A random sample of 1200 engineers included 48 Hispanic Americans,80 African Americans, and 204 females. Determine 90% CI for the proportion of all engineers that are a) female b) Hispanic Americans or African American
To determine the confidence intervals for the proportion of all engineers that are female and Hispanic Americans or African Americans, we can use the formula for calculating confidence intervals for proportions:
CI = p̂ ± Z * sqrt((p̂(1-p̂))/n)
Where:
CI = Confidence Interval
p̂ = Sample proportion
Z = Z-value for the desired confidence level
n = Sample size
a) Female Engineers:
Step 1: Calculate the sample proportion (p̂) of female engineers:
p̂ = Number of females in the sample / Total sample size
p̂ = 204 / 1200
p̂ = 0.17
Step 2: Calculate the Z-value for a 90% confidence level.
For a 90% confidence level, the Z-value is approximately 1.645.
Step 3: Calculate the standard error (SE) of the proportion:
SE = sqrt((p̂(1-p̂))/n)
SE = sqrt((0.17(1-0.17))/1200)
SE = 0.0107 (rounded to four decimal places)
Step 4: Calculate the confidence interval using the formula:
CI = p̂ ± Z * SE
CI = 0.17 ± 1.645 * 0.0107
CI = 0.17 ± 0.0176
CI = (0.1524, 0.1876)
Therefore, the 90% confidence interval for the proportion of all engineers that are female is (0.1524, 0.1876).
b) Hispanic Americans or African Americans:
Step 1: Calculate the sample proportion (p̂) of Hispanic Americans or African Americans:
p̂ = (Number of Hispanic Americans + Number of African Americans) / Total sample size
p̂ = (48 + 80) / 1200
p̂ = 0.107
Step 2: Calculate the Z-value for a 90% confidence level (same as in part a).
Step 3: Calculate the standard error (SE) of the proportion:
SE = sqrt((p̂(1-p̂))/n)
SE = sqrt((0.107(1-0.107))/1200)
SE = 0.0062 (rounded to four decimal places)
Step 4: Calculate the confidence interval using the formula:
CI = p̂ ± Z * SE
CI = 0.107 ± 1.645 * 0.0062
CI = 0.107 ± 0.0102
CI = (0.0968, 0.1172)
Therefore, the 90% confidence interval for the proportion of all engineers that are Hispanic Americans or African Americans is (0.0968, 0.1172).
To determine the confidence intervals (CIs) for proportions, you can use the formula:
CI = p̂ ± Z * √(p̂(1-p̂)/n)
where p̂ is the sample proportion, Z is the Z-score corresponding to the required confidence level, and n is the sample size.
a) To determine the 90% CI for the proportion of all engineers that are female:
Step 1: Calculate the sample proportion (p̂) of females:
p̂ = n_females / n_total = 204 / 1200
Step 2: Find the Z-score corresponding to a 90% confidence level. The formula for the Z-score is:
Z = (1 - confidence level + (confidence level/2))
For a 90% confidence level, Z is approximately 1.645.
Step 3: Calculate the margin of error (ME):
ME = Z * √(p̂(1-p̂)/n_total)
Step 4: Calculate the lower and upper bounds of the confidence interval:
Lower bound = p̂ - ME
Upper bound = p̂ + ME
Plugging in the values:
Lower bound = p̂ - ME = p̂ - (Z * √(p̂(1-p̂)/n_total))
Upper bound = p̂ + ME = p̂ + (Z * √(p̂(1-p̂)/n_total))
b) To determine the 90% CI for the proportion of all engineers that are Hispanic Americans or African Americans:
Step 1: Calculate the sample proportion (p̂) of Hispanic Americans or African Americans:
p̂ = (n_Hispanic + n_African)/ n_total = (48 + 80) / 1200
Step 2: Proceed with steps 2 to 4 as outlined above.
By following these steps, you can determine the confidence intervals for both female engineers and Hispanic American or African American engineers.