91Ó°ÊÓ

Of all freshman at a large college, \(16 \%\) made the dean's list in the current year. As part of a class project, students randomly sample 40 students and check if those students made the list. 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) Calculate the variability of this distribution. (d) What is the formal name of the value you computed in (c)? (e) Suppose the students decide to sample again, this time collecting 90 students per sample, and they again 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 40 observations?

Step by step solution

01

Identify the type of distribution

The distribution formed by taking many samples of the same size from a population, and then plotting a statistic (like the sample proportion) from each sample is called the sampling distribution. (a) The answer is the sampling distribution of the sample proportion.

02

Determine the shape of distribution

Generally, according to the Central Limit Theorem, the sampling distribution of the sample proportion will be approximately symmetric and normal if the sample size is large enough and the success-failure condition is met. The success-failure condition requires that both \( np \) and \( n(1-p) \) are greater than 10.For \( n = 40 \): \( np = 40 \times 0.16 = 6.4 \) \( n(1-p) = 40 \times 0.84 = 33.6 \) Since \( 6.4 \lt 10 \), the condition is not met, so it might be slightly skewed.(b) The answer is that it might be right-skewed due to not meeting the success-failure condition.

03

Calculate the variability

The standard deviation of the sampling distribution of the sample proportion, also known as the standard error, is given by the formula: \[ \sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}} \] Plugging in the values: \( p = 0.16 \), \( n = 40 \), \[ \sigma_{\hat{p}} = \sqrt{\frac{0.16(1-0.16)}{40}} = \sqrt{\frac{0.16 \times 0.84}{40}} \approx 0.058 \](c) The variability is approximately 0.058.

04

Name the computed value

The value computed is known as the standard error of the sample proportion. (d) The formal name of this value is the standard error.

05

Compare variability of new distribution with n=90

For a sample size of 90, we recalculate the standard error: \[ \sigma_{\hat{p}} = \sqrt{\frac{0.16(1-0.16)}{90}} = \sqrt{\frac{0.16 \times 0.84}{90}} \approx 0.038 \] The larger the sample size, the smaller the standard error, indicating less variability.(e) The variability will be smaller when the sample size is 90 compared to 40, indicating the new distribution is more concentrated around the mean.

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Key Concepts

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

Central Limit Theorem

When we talk about the Central Limit Theorem (CLT), we're dealing with a fundamental principle in statistics. What the theorem essentially tells us is that if you have a large enough sample size, the distribution of the sample means will be approximately normal, even if the original data distribution isn't normal. This is incredibly useful because it allows us to make inferences about populations.

Some key insights include:

• The sample size really matters. The larger the sample, the more the distribution of the sample mean resembles a normal distribution.
• However, if the original population is not normal, a larger sample size is necessary to achieve that normality.
• This theorem is applicable no matter the shape of the population distribution, but usually, a sample size of more than 30 is considered sufficient.

In the context of our exercise, the Central Limit Theorem explains the shape of the distribution of the sample proportions. When n = 40, the CLT does not hold perfectly because the condition of np > 10 and n(1-p) > 10 is not met, leading to a slight skew rather than perfect symmetry.

Standard Error

The standard error is a measure of how much variation or "spread" we can expect in the sample means or proportions if we were to take multiple samples from the same population. It's especially crucial in understanding how the observed sample means differ from the true population mean.

Here's why it's important:

• It quantifies the variability of a sample statistic (like a sample mean or proportion) as an estimate of the population parameter.
• A smaller standard error indicates that our sample mean is a more precise estimate of the true population mean.
• The standard error decreases as the sample size increases because the estimate becomes more accurate.

In relation to the problem at hand, we calculated the standard error of the sample proportion for n = 40 and n = 90. For n = 40, the standard error was approximately 0.058, while for n = 90, it was approximately 0.038. This shows the principle that a larger sample size results in a smaller standard error, meaning more reliable results.

Sample Proportion

A sample proportion refers to the fraction of samples that have a particular characteristic, which in this case is making the dean's list. Determining the sample proportion helps us estimate the likelihood or frequency of an event happening within our sample data.

Let's break it down:

• The sample proportion is calculated by dividing the number of successes by the total number of observations in the sample.
• It serves as a point estimate for the population proportion, providing a snapshot of the population from which the sample was drawn.
• The accuracy of the sample proportion estimate improves with a larger sample size, making it more representative of the true population.

In the example given, each sample represents the proportion of students who made the dean's list. By aggregating 1,000 samples, the students generated a sampling distribution of sample proportions, which is a powerful tool to visualize and analyze the data distribution and variability. This technique allows for generalizations about the population based on the proportion of interest.