Question

Calculate \(\operatorname{SE}(\hat{p})\) for \(n=100\) and these values of \(p\) : a. \(p=.01\) b. \(p=.10\) c. \(p=.30\) d. \(p=.50\) e. \(p=.70\) f. \(p=.90\) g. \(p=.99\) h. Plot \(\operatorname{SE}(\hat{p})\) versus \(p\) on graph paper and sketch a smooth curve through the points. For what value of \(p\) is the standard deviation of the sampling distribution of \(\hat{p}\) a maximum? What happens to the standard error when \(p\) is near 0 or near \(1.0 ?\)

Short answer

Answer: The maximum standard deviation of the sampling distribution occurs when 𝑝 = 0.50. When 𝑝 is close to 0 or 1, the standard error tends to decrease towards zero, showing that the variation in the sampling distribution is minimized.

Step-by-step solution

Calculating Standard Error

Follow the formula, \(\operatorname{SE}(\hat{p}) = \sqrt{\frac{p(1-p)}{n}}\), and replace \(n\) with 100. Compute the standard error for each value of \(p\) provided: a. \(p = .01\): \(\operatorname{SE}(\hat{p}) = \sqrt{\frac{(.01)(1 - .01)}{100}}\) b. \(p = .10\): \(\operatorname{SE}(\hat{p}) = \sqrt{\frac{(.10)(1 - .10)}{100}}\) c. \(p = .30\): \(\operatorname{SE}(\hat{p}) = \sqrt{\frac{(.30)(1 - .30)}{100}}\) d. \(p = .50\): \(\operatorname{SE}(\hat{p}) = \sqrt{\frac{(.50)(1 - .50)}{100}}\) e. \(p = .70\): \(\operatorname{SE}(\hat{p}) = \sqrt{\frac{(.70)(1 - .70)}{100}}\) f. \(p = .90\): \(\operatorname{SE}(\hat{p}) = \sqrt{\frac{(.90)(1 - .90)}{100}}\) g. \(p = .99\): \(\operatorname{SE}(\hat{p}) = \sqrt{\frac{(.99)(1 - .99)}{100}}\)

Plotting Standard Error vs. Population Proportion

Plot all the calculated standard error values against \(p\) on graph paper. This plot will show how the standard error (\(\operatorname{SE}(\hat{p})\)) changes as the population proportion (\(p\)) varies. Draw a smooth curve through the points.

Analyzing the Graph

Observe the smooth curve passing through the plotted points: 1. Identify the value of \(p\) for which the standard deviation of the sampling distribution of \(\hat{p}\) is the maximum. 2. Observe the behavior of the standard error when \(p\) is near 0 or near 1.0. Based on the graph, it can be concluded that the maximum standard deviation occurs when \(p = 0.50\). This is because the curve will be symmetrical around \(p=0.5\), and the graph would have the shape of an inverted parabola. When \(p\) is near 0 or near 1.0, the standard error tends to decrease towards zero, as the curve touches the x-axis at these points. This shows that the variation in the sampling distribution is minimized when the population proportion is very small or very close to 1.

Key concepts

Sampling Distribution

In statistics, when we talk about sampling distribution, we're referring to the probability distribution of a statistic, like a sample mean or sample proportion, obtained from a large number of samples drawn from a specific population. Imagine repeatedly taking random samples from a population: the distribution of the sample proportions from these samples forms the sampling distribution of the proportion estimate. It provides insights into the variability and reliability of the sample statistic as an estimator of the population parameter.
For example, if we take samples from a population where a characteristic occurs with a certain probability (known as the population proportion), the sampling distribution shows us the pattern in which the proportions of this characteristic occur among the different samples. Knowing the shape and spread of this distribution helps in making accurate inferences about the population from the data.

Proportion Estimate

A proportion estimate, denoted as \(\hat{p}\), refers to the sample proportion or the ratio obtained by dividing the number of items with a specific characteristic by the total number of items in the sample. This is essential when trying to infer the population proportion (denoted as \(p\)) from a sample. It's especially useful in surveys, polls, and experiments.
For example, if you're trying to estimate the proportion of voters favoring a new policy, the proportion of your sample who support the policy (based on survey responses) provides a proportion estimate. Accurate estimation is crucial, as it determines how well your survey or experiment can infer the true population proportion.

Standard Deviation

Standard Deviation is a measure of the amount of variation or dispersion of a set of values. In terms of sampling distributions, it indicates how much the sample proportions deviate from the population proportion. When speaking about the standard error of the proportion estimate, it's understood as the standard deviation of the sampling distribution of the sample proportion. This is calculated using the formula \(\text{SE}(\hat{p}) = \sqrt{\frac{p(1-p)}{n}}\), where \(n\) is the sample size.
The standard deviation helps to understand how sample results might vary if the sampling were repeated multiple times. A small standard deviation means the data points tend to be closer to the mean value of the set, indicating a reliable estimate of the population proportion.

Population Proportion

Population Proportion, often symbolized as \(p\), is the ratio of members of a population that have a certain attribute to the total members of the population. This is a fundamental parameter in inferential statistics, as it is often unknown and needs to be estimated from sample data.
For instance, in a city, if you're estimating how many people prefer public transport, the population proportion is the fraction of the total population who prefer it. Since this real value is typically unknown, statisticians use sample proportions to infer it. Accurate estimation of \(p\) is vital for making correct decisions and predictions about the population's behavior and preferences.

Graphical Analysis

Graphical Analysis involves visually interpreting data using graphs and charts to detect patterns, trends, and outliers. For statistical topics like standard error, graphical representation helps to understand how estimates behave as variables like sample size or population proportion change.
For instance, plotting the standard error against various population proportions shows how the variability of \(\hat{p}\) changes with different \(p\) values. This graph typically forms an inverted parabola around \(p=0.5\), indicating maximum standard deviation. When \(p\) is near 0 or 1, the standard error approaches zero, signifying that sample proportions are less variable and more predictable than those in the middle range of \(p\). Understanding these patterns is crucial for making informed interpretations of statistical data.