91Ó°ÊÓ

A nonprofit wants to understand the fraction of households that have elevated levels of lead in their drinking water. They expect at least \(5 \%\) of homes will have elevated levels of lead, but not more than about \(30 \%\). They randomly sample 800 homes and work with the owners to retrieve water samples, and they compute the fraction of these homes with elevated lead levels. They repeat this 1,000 times and build a distribution of sample proportions. (a) What is this distribution called? (b) Would you expect the shape of this distribution to be symmetric, right skewed, or left skewed? Explain your reasoning. (c) If the proportions are distributed around \(8 \%\), what is the variability of the distribution? (d) What is the formal name of the value you computed in (c)? (e) Suppose the researchers' budget is reduced, and they are only able to collect 250 observations per sample, but they can still collect 1,000 samples. They build a new distribution of sample proportions. How will the variability of this new distribution compare to the variability of the distribution when each sample contained 800 observations?

Step by step solution

01

Understanding the distribution

The distribution described in the exercise is the sampling distribution of the sample proportion. This is because it is built by taking repeated random samples (1,000 samples) of a given size (800 homes) and then computing the sample proportion of homes with elevated lead levels in each sample.

02

Analyzing the shape of the distribution

Since the sample size is large (n = 800), the distribution is expected to be approximately normal due to the Central Limit Theorem, which says that sample proportions will be approximately normally distributed if the sample size is large enough. Thus, we expect the distribution to be symmetric.

03

Calculating variability of the distribution

To find the variability of the distribution when the sample proportion is around 8%, use the formula for the standard error of the sample proportion: \[SE = \sqrt{\frac{p(1-p)}{n}}\]where \(p = 0.08\) and \(n = 800\). \[SE = \sqrt{\frac{0.08 \times 0.92}{800}} \approx 0.0096\]

04

Identifying the formal name for variability

The formal name of the value computed in Step 3 is "Standard Error" of the sample proportion. It measures the variability of the sample proportion across different samples.

05

Comparing variability with reduced sample size

With a reduced sample size of 250, the standard error of the sample proportion increases, calculated as:\[SE = \sqrt{\frac{0.08 \times 0.92}{250}} \approx 0.0172\]Thus, the variability of the distribution with 250 observations is larger than when the sample size was 800.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Central Limit Theorem

The Central Limit Theorem (CLT) is a fundamental principle in the realm of statistics. It allows us to use sample data to make inferences about a population. The theorem states that when you take a sufficiently large number of samples from a population, each with the same size, the distribution of the sample means will approach a normal distribution, regardless of the original distribution of the population.

One of the keys to understanding the CLT is recognizing that the sample size plays a crucial role. Typically, a sample size of 30 or more is considered 'sufficiently large' for the CLT to apply. However, when dealing with proportions, this 'sufficiently large' condition might require larger samples. As seen in the provided exercise, with a sample size of 800, the shape of the sampling distribution becomes approximately normal, allowing us to use standard statistical tools to analyze it.

• Large sample sizes lead to more accurate approximations of normally distributed sample means.
• The CLT enables estimation and hypothesis testing concerning population parameters.

Standard Error

The standard error is a key concept in the context of sampling distributions. It provides a measure of the amount of variation or dispersion of the sample statistic from the population parameter. Specifically, in the context of proportions, the standard error indicates the variability of the sample proportion from sample to sample. This helps in understanding how much the sample proportion can vary as sampling continues.

Standard error is not to be confused with standard deviation, which measures variability within a single sample. Instead, standard error focuses on variability between multiple samples. It is calculated using the formula:

\[ SE = \sqrt{\frac{p(1-p)}{n}} \]
Where:

• \( p \) is the sample proportion.
• \( n \) is the sample size.

As the exercise illustrates, with a sample proportion of \(0.08\) and a sample size of \(800\), the standard error comes out to \(0.0096\). This tells us how much we might expect the proportion of homes with elevated lead levels to vary across the different samples taken.

Sample Proportion

The sample proportion is a statistic that represents the fraction or percentage of the sample that meets a specific criterion. In the exercise's context, the sample proportion refers to the fraction of homes in each sample with elevated lead levels.

Understanding the sample proportion is critical because it serves as an estimate or representation of the true proportion of the population. When multiple samples are taken, as in the 1,000 samples collected in the exercise, the sample proportions form a distribution called the "sampling distribution of the sample proportion." This distribution can then be analyzed for patterns and variability using concepts like the standard error.

• Sample proportion helps to estimate population parameters.
• It is instrumental in forming confidence intervals and hypothesis testing regarding the population proportion.

Variance

Variance is a measure of how much values in a data set differ from the mean of the data set. In the context of sampling distributions, variance offers insight into the spread of the sample mean or proportion.

Calculating the variance of a sample proportion involves using the binomial model because proportions derive from binary outcomes - success or failure, such as homes with or without elevated lead levels. With the sample proportion denoted as \( p \), the variance \( \sigma^2 \) is calculated as:

\[ \sigma^2 = \frac{p(1-p)}{n} \]
Where \( n \) is the sample size.

• Variance is essential for understanding data dispersion.
• It reflects the degree to which data points differ from the mean.

In the exercise, calculating the variance aids in understanding how much we expect the sample proportion to vary based on the proportion and sample size used for each example of sampling within the 1,000 iterations.