Researchers are estimating a population proportion using a sampling distribution of sample proportions with sample size $$ n $$. If the researchers want to increase 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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To determine how to increase 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 $$ n $$**: This would decrease the standard error because $$ n $$ is in the denominator of the fraction.

2. **Decreasing the sample size $$ n $$**: This would increase the standard error because a smaller sample size leads to more variability in the estimate.

3. **Increasing the sample proportion $$ \hat{p} $$**: This affects the term $$ \hat{p}(1 - \hat{p}) $$. The maximum value of this term occurs when $$ \hat{p} = 0.5 $$, which gives the highest variability. If $$ \hat{p} $$ is increased beyond this point, $$ \hat{p}(1 - \hat{p}) $$ will decrease, leading to a smaller standard error.

4. **Decreasing the sample proportion $$ \hat{p} $$**: Similar to increasing $$ \hat{p} $$, if $$ \hat{p} $$ is decreased, it can also lead to a decrease in variability, depending on how far it is from 0.5.

Given this analysis, the correct action to **increase the standard error** of the sampling distribution is to:

**decrease the sample size**.
To increase the standard error of the sampling distribution, the researchers should **decrease the sample size**.

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