An SRS of size 100 is taken from a population having 80% or a proportion of 0.80 successes An independent SRS of size 400 is taken from a population having 50% or a proportion of 0.50 successes. The standarderror used for a CONFIDENCE INTERVAL for the difference in population proportions is: 0.40 0.047 013 0.002 0.00005
The Correct Answer and Explanation is:
To compute the standard error (SE) for the difference in population proportions, we use the formula for the standard error of the difference between two independent proportions: SE=p1(1−p1)n1+p2(1−p2)n2SE = \sqrt{ \frac{p_1 (1 – p_1)}{n_1} + \frac{p_2 (1 – p_2)}{n_2} }SE=n1p1(1−p1)+n2p2(1−p2)
Where:
- p1=0.80p_1 = 0.80p1=0.80, n1=100n_1 = 100n1=100
- p2=0.50p_2 = 0.50p2=0.50, n2=400n_2 = 400n2=400
Step-by-step Calculation
SE=0.80×(1−0.80)100+0.50×(1−0.50)400SE = \sqrt{ \frac{0.80 \times (1 – 0.80)}{100} + \frac{0.50 \times (1 – 0.50)}{400} }SE=1000.80×(1−0.80)+4000.50×(1−0.50) SE=0.80×0.20100+0.50×0.50400SE = \sqrt{ \frac{0.80 \times 0.20}{100} + \frac{0.50 \times 0.50}{400} }SE=1000.80×0.20+4000.50×0.50 SE=0.16100+0.25400SE = \sqrt{ \frac{0.16}{100} + \frac{0.25}{400} }SE=1000.16+4000.25 SE=0.0016+0.000625=0.002225SE = \sqrt{0.0016 + 0.000625} = \sqrt{0.002225}SE=0.0016+0.000625=0.002225 SE≈0.0472SE \approx 0.0472SE≈0.0472
Correct Answer: 0.047
Explanation
In statistics, when comparing two population proportions based on independent samples, we use the standard error to measure the variability in the difference between the sample proportions. The formula for the standard error of the difference between two independent sample proportions accounts for the variability in each individual sample. It is derived from the properties of variance for independent random variables.
In this case, we are given two samples. The first sample has 100 observations and a population proportion of success of 0.80. The second has 400 observations and a population proportion of success of 0.50. Since these are independent samples, we can compute their individual variances and then sum them to find the variance of the difference.
The standard error is essentially the square root of that total variance. This tells us how much we expect the sample difference in proportions to vary due to random sampling. A smaller standard error means more precision in our estimate of the difference between population proportions.
The calculated standard error of approximately 0.047 means that if we were to repeatedly sample from these populations in the same way, the differences in sample proportions would typically vary by about 0.047 from the true population difference. This value is then used in constructing confidence intervals. For instance, a 95 percent confidence interval would involve taking the observed difference in sample proportions and adding and subtracting a multiple of the standard error to create a range of plausible values for the true population difference.
Thus, the correct choice from the options given is 0.047.
