The standard deviation of all possible sample proportions increases as the sample size increases - TRUE OR FALSE
As the sample size increases, the standard deviation decreases.
TRUE.
To explain why, we first need to understand what a sample proportion and standard deviation are.
A sample proportion represents the proportion of a specific outcome in a sample, usually expressed as a decimal or percentage. For example, if we are interested in the proportion of individuals who prefer coffee over tea, the sample proportion would represent the percentage of individuals in our sample who prefer coffee.
Standard deviation, on the other hand, measures the variability or spread of a set of values. In this case, it represents the variability of sample proportions.
As the sample size increases, the standard deviation of the sample proportions also increases. This is known as the "standard error of the proportion" or "sampling error."
To understand why this happens, we can consider the formula for calculating the standard deviation of sample proportions:
Standard deviation of sample proportions = √[(p * (1 - p)) / n]
Where:
- p represents the true population proportion
- n represents the sample size
Notice that as the sample size (n) gets larger, the denominator of the formula becomes larger. This leads to a smaller standard deviation. Conversely, as the sample size (n) gets smaller, the denominator becomes smaller, leading to a larger standard deviation.
In other words, as the sample size increases, the variability of the sample proportions decreases, resulting in a smaller standard deviation. This suggests that larger sample sizes provide more reliable estimates of the population proportion.
Therefore, the statement "The standard deviation of all possible sample proportions increases as the sample size increases" is FALSE. The standard deviation actually decreases as the sample size increases.