A random sample of 384 people in Dalto City, a mid-sized city, revealed 112 individuals who work at more than one job. A second random sample of 432 workers from East Dettweiler, another mid-sized city, found 91 people who work at more than one job. To conduct a significance test for a difference in the proportions of workers in Dalto City and East Dettweiler who work at more than one job, what's the pooled value for p̂, and what's the pooled standard error of the difference between the two sample proportions, respectively?
.5, .0303
.249, .0009
.249, .017
.251, .0303
.249, .0303
To find the pooled value for p̂, we can calculate the overall proportion of workers who work at more than one job across both cities.
For Dalto City:
Sample size (n1) = 384
Number of individuals who work at more than one job (x1) = 112
p̂1 = x1 / n1 = 112 / 384 ≈ 0.2917
For East Dettweiler:
Sample size (n2) = 432
Number of individuals who work at more than one job (x2) = 91
p̂2 = x2 / n2 = 91 / 432 ≈ 0.2106
The pooled value for p̂ is the overall proportion of workers who work at more than one job:
p̂ = (x1 + x2) / (n1 + n2) = (112 + 91) / (384 + 432) ≈ 0.249
Therefore, the pooled value for p̂ is approximately 0.249.
To find the pooled standard error of the difference between the two sample proportions, we can use the formula:
SE(p̂1 - p̂2) = sqrt[(p̂ * (1 - p̂) * (1/n1 + 1/n2))]
SE(p̂1 - p̂2) = sqrt[(0.249 * (1 - 0.249) * (1/384 + 1/432))]
SE(p̂1 - p̂2) ≈ 0.0303
Therefore, the pooled standard error of the difference between the two sample proportions is approximately 0.0303.
Hence, the correct answer is:
.249, .0303.
To find the pooled value for p-hat (p̂) and the pooled standard error of the difference between the two sample proportions, you need to follow these steps:
Step 1: Calculate the proportion of workers who work at more than one job in each city.
For Dalto City:
p̂1 = Number of individuals who work at more than one job in Dalto City / Total sample size in Dalto City
= 112 / 384
≈ 0.2917
For East Dettweiler:
p̂2 = Number of individuals who work at more than one job in East Dettweiler / Total sample size in East Dettweiler
= 91 / 432
≈ 0.2106
Step 2: Calculate the pooled value for p-hat (p̂).
To calculate the pooled value for p-hat (p̂), use the formula:
p̂ = (p̂1 * n1 + p̂2 * n2) / (n1 + n2)
where p̂1 and p̂2 are the proportions and n1 and n2 are the sample sizes.
Using the given values:
p̂ = (0.2917 * 384 + 0.2106 * 432) / (384 + 432)
≈ 0.249
Therefore, the pooled value for p̂ is approximately 0.249.
Step 3: Calculate the pooled standard error of the difference between the two sample proportions.
To calculate the pooled standard error, use the formula:
pooled standard error = √[(p̂ * (1 - p̂)) / n1 + (p̂ * (1 - p̂)) / n2]
Using the given values:
pooled standard error = √[(0.249 * (1 - 0.249)) / 384 + (0.249 * (1 - 0.249)) / 432]
≈ √[0.186 / 384 + 0.186 / 432]
≈ √(0.000484 + 0.000431)
≈ √0.000915
≈ 0.0303
Therefore, the pooled standard error of the difference between the two sample proportions is approximately 0.0303.
Hence, the correct answer is:
.249, .0303