Researchers are estimating a population proportion using a sampling distribution of sample proportions with sample size $$ n $$. If the researchers want to decrease the standard error of their sampling distribution they should... increase the sample size decrease the sample size increase the sample proportion $$ \hat{p} $$ decrease the sample proportion $$ \hat{p} $$

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UpStudy AI · Accepted Answer

To determine how to decrease the standard error of the sampling distribution of sample proportions, we need to understand the formula for the standard error (SE) of the sample proportion $$ \hat{p} $$:

$$
SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}
$$

Where:
- $$ \hat{p} $$ is the sample proportion,
- $$ n $$ is the sample size.

From this formula, we can see that the standard error is influenced by both the sample size $$ n $$ and the sample proportion $$ \hat{p} $$.

1. **Increasing the sample size $$ n $$**: 
   - If we increase $$ n $$, the denominator of the fraction under the square root increases, which results in a smaller standard error. Therefore, increasing the sample size decreases the standard error.

2. **Decreasing the sample size $$ n $$**: 
   - If we decrease $$ n $$, the denominator decreases, leading to a larger standard error. Thus, decreasing the sample size increases the standard error.

3. **Increasing the sample proportion $$ \hat{p} $$**: 
   - The term $$ \hat{p}(1 - \hat{p}) $$ reaches its maximum when $$ \hat{p} = 0.5 $$. If $$ \hat{p} $$ is increased beyond this point, $$ (1 - \hat{p}) $$ decreases, which can lead to a decrease in the product $$ \hat{p}(1 - \hat{p}) $$. However, this effect is not as straightforward as changing $$ n $$.

4. **Decreasing the sample proportion $$ \hat{p} $$**: 
   - Similar to increasing $$ \hat{p} $$, decreasing $$ \hat{p} $$ can also affect the product $$ \hat{p}(1 - \hat{p}) $$, but again, the effect is not as significant as changing $$ n $$.

### Conclusion:
To decrease the standard error of the sampling distribution, the researchers should **increase the sample size** $$ n $$.

Thus, the correct answer is:
- **Increase the sample size**.
To decrease the standard error, increase the sample size.

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