Question

Is it appropriate to use the normal distribution to approximate the sampling distribution of \(\hat{p}\) in the following circumstances? a. \(n=50, p=.05\) b. \(n=75, p=.1\) c. \(n=250, p=.99\)

Short answer

a. \(n=50, p=.05\) b. \(n=75, p=.1\) c. \(n=250, p=.99\) Answer: It is not appropriate to use the normal distribution to approximate the sampling distribution of \(\hat{p}\) in any of the given circumstances.

Step-by-step solution

a. \(n=50, p=.05\)

First, we need to check if the sample size is large enough. Calculate \(np\) and \(n(1-p)\): \(np = 50 \times 0.05 = 2.5\) \(n(1-p) = 50 \times (1-0.05) = 47.5\) Since \(np\) is less than 10, it is not appropriate to use the normal distribution to approximate the sampling distribution of \(\hat{p}\) in this case.

b. \(n=75, p=.1\)

First, we need to check if the sample size is large enough. Calculate \(np\) and \(n(1-p)\): \(np = 75 \times 0.1 = 7.5\) \(n(1-p) = 75 \times (1-0.1) = 67.5\) Since \(np\) is less than 10, it is not appropriate to use the normal distribution to approximate the sampling distribution of \(\hat{p}\) in this case.

c. \(n=250, p=.99\)

First, we need to check if the sample size is large enough. Calculate \(np\) and \(n(1-p)\): \(np = 250 \times 0.99 = 247.5\) \(n(1-p) = 250 \times (1-0.99) = 2.5\) Since \(n(1-p)\) is less than 10, it is not appropriate to use the normal distribution to approximate the sampling distribution of \(\hat{p}\) in this case. In conclusion, it is not appropriate to use the normal distribution to approximate the sampling distribution of \(\hat{p}\) in any of the given circumstances.

Key concepts

Sampling Distribution

The concept of sampling distribution is key in statistics. It refers to the probability distribution of a statistic obtained through a large number of samples drawn from a specific population. Each sample consists of a certain number of observations, and its statistics, like the sample mean or sample proportion, can differ from sample to sample. When you repeat the sampling process many times, you'll have a distribution of those statistics which we call the sampling distribution.
This distribution can help us understand how the sample statistic is likely to vary and whether it accurately reflects the population parameter. A common task in statistics is to determine the shape of this distribution. Under specific conditions, it might resemble a normal distribution, which eases further analysis and calculations.
It's important to assess whether a normal distribution can approximate a sampling distribution based on criteria like sample size and proportion values.

Sample Size

Sample size plays a crucial role in determining the appropriateness of using a normal distribution for approximation. Large samples generally lead to more reliable and stable estimates of the population parameter. In statistics, a rule of thumb is to ensure that both the product of the sample size () and the sample proportion (p), and its complement (1-p), should be at least 10.
This condition: \(np \geq 10\) and \(n(1-p) \geq 10\) is used to justify the use of the normal distribution as an approximation of the sampling distribution. A larger sample size also reduces the margin of error, leading to more precise estimates. If p or (1-p) is small, the sample would need to be much larger for the approximation to be valid.

Approximation

The process of approximation involves using a known distribution to estimate a sampling distribution when it's complicated or inconvenient to determine its exact form. The normal distribution is frequently used for this purpose due to its well-defined properties. Approximating a sampling distribution with a normal distribution allows statisticians to use z-scores and standard deviation to calculate probabilities quickly and easily.
However, the approximation is valid only under certain conditions, such as having a sufficiently large sample size and certain distribution properties. You should always check the conditions (e.g., \(np \geq 10\) and \(n(1-p) \geq 10\)) to ensure the approximation gives meaningful results.
Otherwise, relying on the normal distribution might lead to inaccurate conclusions about the probability and characteristics of the sample statistics.

Probability

Probability underlies the entire process of inferring from a sample to a population. It's the mathematical framework that allows us to estimate and predict how likely certain outcomes are, given certain conditions. In the context of sampling distributions, probability helps determine whether using a normal distribution approximation is logical.
If conditions are met, the normal distribution can reliably reflect the probability behavior of sample statistics. This is based on the Central Limit Theorem, which states that, for sufficiently large samples, the distribution of the sample mean will be approximately normal regardless of the shape of the population distribution.

• When analyzing sample proportions, calculating the probability that these proportions align closely with population proportions is often of interest.
• Proper probability models allow statisticians to understand the likelihood of various outcomes when drawing samples from a population.

In conclusion, probability allows us to leverage the normal distribution's properties to make predictions based on samples with confidence, only if the conditions for approximation are valid.