Question

Suppose that \[60\%\] of adults in district a support a new ballot measure, while \[45\%\] of adults in district b support the same measure. pollsters take an srs of \[200\] adults from district a and a separate srs of \[100\] adults from district b to see the difference between the sample proportions \[(\hat{p}_\text{a}-\hat{p}_\text{b})\]. what are the mean and standard deviation of the sampling distribution of \[\hat{p}_\text{a}-\hat{p}_\text{b}\]? choose 1 answer: choose 1 answer:

Answer 1

The mean of the sampling distribution of p_a-p_b is 0.15 and the standard deviation is 0.0606.

Step-by-step explanation:

To find the mean and standard deviation of the sampling distribution of ⁢⁡p⁢⁡a-⁢⁡p⁢⁡b, we can use the formula:
Mean: ⁢⁡p⁢⁡a-⁢⁡p⁢⁡b = pa - pb = 0.60 - 0.45 = 0.15
Standard Deviation: ⁢⁡σ⁢⁡p⁢⁡a-⁢⁡p⁢⁡b = sqrt[(pa(1-pa)/na) + (pb(1-pb)/nb)] = sqrt[(0.60(0.40)/200) + (0.45(0.55)/100)] = sqrt[(0.24/200) + (0.2475/100)] = sqrt[0.0012 + 0.002475] = sqrt(0.003675) = 0.0606

Answer 2

The sampling distribution of the difference between two sample proportions (PA - PB) will be approximate normal due to the Central Limit Theorem and since both sample sizes are large.

The shape of the sampling distribution of №(PA - PB) will follow an approximate normal distribution. This conclusion is based on the Central Limit Theorem which asserts that the distribution of the difference between two sample proportions will be approximately normal, provided that the sample sizes are sufficiently large and the conditions for the sampling distribution of proportions are met.

Here, we have an independent simple random sample (SRS) of 200 adults from district A and 100 adults from district B. The sampling distribution of PA - PB is centered at the difference between the population proportions (PA - PB), and its standard deviation can be calculated using the formula for the standard deviation of the difference between two independent proportions. Since both sample sizes are large (nA = 200 > 30 and nB = 100 > 30), we can use the normal approximation for the sampling distribution.

Complete question:

Suppose that 60% of adults in district A support a new ballot measure, while 45% of adults in district B support the same measure. Polisters take an SRS of 200 adults from district A and a separate SRS of 100 adults from district B to see the difference between the sample proportions (PA-PB).

What will be the shape of the sampling distribution of PA - PB, and why?