Question

For two populations of rabbits, R and S, the proportions of rabbits with white markings on their fur are given as pR and pS, respectively. Suppose that independent random samples of 50 rabbits from R and 100 rabbits from S are selected. Let pˆR be the sample proportion of rabbits with white markings from R, and let pˆS be the sample proportion of rabbits with white markings from S. What is the standard deviation of the sampling distribution of pˆR−pˆS ?

Answer 1

The standard deviation of the sampling distribution of the difference between two sample proportions is calculated using a formula involving the population proportions and sample sizes. Without the actual population proportions, we cannot give a numerical value, only the formula in terms of pR and pS.

Step-by-step explanation:

The student's question pertains to the standard deviation of the sampling distribution of the difference between two sample proportions, ⃦R - ⃦S. To find this value, we make use of the formula for the standard deviation of the difference between two independent proportions, which is √[(pR(1-pR)/nR) + (pS(1-pS)/nS)]. The standard deviation can be approximated by the normal distribution due to the Central Limit Theorem for proportions, provided that both np and n(1-p) are greater than five.

In this scenario, we are given that the sample sizes are nR = 50 for population R and nS = 100 for population S. However, since the actual population proportions pR and pS are not provided, we cannot calculate a numerical value. The final answer for the standard deviation of the sampling distribution of ⃦R - ⃦S would be in terms of pR and pS.

Answer 2

The standard deviation is
`\hat p_(R) -\hat p_(S) = \sqrt{(\hat p_(R)(1-\hat p_(R)))/(50) + (\hat p_(S)(1-\hat p_(S)))/(100) }`

Step-by-step explanation:

The standard deviation of a sampling distribution is the standard error or a valuation of the standard deviation. Where statistic parameter is the mean it is referred to as the standard error of the mean.

The formula for standard deviation of a sampling distribution is as follows;

`\hat p_(R) -\hat p_(S) = \sqrt{(\hat p_(R)(1-\hat p_(R)))/(n_R) + (\hat p_(S)(1-\hat p_(S)))/(n_S) }`

Where;

`{\hat p_(R)}` = Sample proportion of rabbits with white markings from R

`{\hat p_(S)}` = Sample proportion of rabbits with white markings from S

`n_R` = Number of from R = 50

`n_S` = Number of from S = 100

Therefore, the standard deviation of the sampling distribution is given as follows;

`\hat p_(R) -\hat p_(S) = \sqrt{(\hat p_(R)(1-\hat p_(R)))/(50) + (\hat p_(S)(1-\hat p_(S)))/(100) }`.