An SRS of size 100 is taken from a population having a proportion of 0.8 of successes. An independent SRS of size 400 is taken from a population having a proportion of 0.5 of successes. The sampling distribution for the difference in the sample proportions p1 – p2 has a standard deviation equal to?
The Correct Answer and Explanation is:
To calculate the standard deviation of the sampling distribution of the difference in sample proportions p1−p2p_1 – p_2p1−p2, we use the formula:Standard deviation=p1(1−p1)n1+p2(1−p2)n2\text{Standard deviation} = \sqrt{\frac{p_1(1 – p_1)}{n_1} + \frac{p_2(1 – p_2)}{n_2}}Standard deviation=n1p1(1−p1)+n2p2(1−p2)
Where:
- p1=0.8p_1 = 0.8p1=0.8, n1=100n_1 = 100n1=100
- p2=0.5p_2 = 0.5p2=0.5, n2=400n_2 = 400n2=400
Substituting into the formula:Standard deviation=0.8(1−0.8)100+0.5(1−0.5)400=0.8×0.2100+0.5×0.5400=0.16100+0.25400=0.0016+0.000625=0.002225≈0.0472\text{Standard deviation} = \sqrt{\frac{0.8(1 – 0.8)}{100} + \frac{0.5(1 – 0.5)}{400}} = \sqrt{\frac{0.8 \times 0.2}{100} + \frac{0.5 \times 0.5}{400}} = \sqrt{\frac{0.16}{100} + \frac{0.25}{400}} = \sqrt{0.0016 + 0.000625} = \sqrt{0.002225} \approx 0.0472Standard deviation=1000.8(1−0.8)+4000.5(1−0.5)=1000.8×0.2+4000.5×0.5=1000.16+4000.25=0.0016+0.000625=0.002225≈0.0472
Correct Answer: Approximately 0.0472
Explanation
The sampling distribution of the difference in sample proportions is used to estimate how much the observed difference between two sample proportions would vary if the sampling were repeated multiple times. This concept is essential in hypothesis testing and confidence interval estimation when comparing proportions from two different populations.
In this case, we are dealing with two independent simple random samples. The first sample has a proportion of successes equal to 0.8 from a sample size of 100. The second sample has a proportion of 0.5 from a larger sample of size 400. Because the samples are independent, the variances of the sample proportions can be added to find the overall variability in the difference.
The variability of a single sample proportion is calculated using p(1−p)/np(1 – p) / np(1−p)/n. Since we are dealing with two proportions, the standard deviation of the difference combines both variances under a square root. This gives a sense of how much the observed difference between the two sample proportions might vary due to sampling error alone.
In practical terms, the resulting standard deviation of approximately 0.0472 tells us that if we repeatedly sampled from these two populations, the difference in sample proportions would vary around the true difference with a spread of about 0.0472. This value is especially useful when constructing a confidence interval or conducting a significance test to determine whether the difference in population proportions is statistically meaningful.
