Question

When they send out their fundraising letter, a philanthropic organization typically gets a return from about 5% of the people on their mailing list. To see what the response rate might be for future appeals, they did a simulation using samples of size 20, 50, 100, and 200. For each sample size, they simulated 1000 mailings with success rate p = 0.05 and constructed the histogram of the 1000 sample proportions. Explain how these four histograms demonstrate what the Central Limit Theorem says about the sampling distribution model for sample proportions. Be sure to talk about shape, center, and spread.

Answer 1

The histograms demonstrate the central limit theorem for sample proportions in terms of shape, center, and spread.

Step-by-step explanation:

The four histograms demonstrate what the Central Limit Theorem says about the sampling distribution model for sample proportions. According to the Central Limit Theorem, as the sample size increases, the sampling distribution of sample proportions becomes approximately normal.

Shape: As the sample size increases, the shape of the histograms becomes more symmetric and bell-shaped, resembling a normal distribution.

Center: The center of the histograms remains approximately equal to the true population proportion of 0.05.

Spread: The spread of the histograms, or the variability of the sample proportions, decreases as the sample size increases.

Answer 2

The histograms in the philanthropic organization's simulation exemplify the Central Limit Theorem by showing that as sample sizes increase, the distribution of sample proportions approaches a normal distribution with the center at the population proportion and spread decreasing with larger sample sizes.

Step-by-step explanation:

The philanthropic organization is dealing with a proportion problem because they are collecting categorical data; specifically, they are measuring the proportion of respondents who reply to their fundraising letter with a success (yes) or failure (no). This type of data analysis aids in estimating the true population proportion on various criteria such as smoking rates or voting preferences.

According to the Central Limit Theorem (CLT) for proportions, as the sample size n increases, the distribution of the sample proportions (P') will approximate a normal distribution with a mean equal to the true population proportion p and standard deviation equal to √(pq/n), where q is 1-p. In the histograms constructed based on different sample sizes (20, 50, 100, 200) for the simulation of 1000 mailings, we would expect the following characteristics: as the sample size increases, the histograms will look more normally distributed (bell-shaped); the center of the histograms will be close to the true proportion of responses (p = 0.05); and the spread (or variability) of the histograms will decrease with larger sample sizes, making the histogram narrower as specified by the decreasing value of the standard deviation √(pq/n).

Thus, these histograms provide a visual representation of the CLT, demonstrating that for large enough sample sizes, the sampling distribution of the sample proportion P' becomes normally distributed, centered around the true population proportion p, and with a spread inversely related to the square root of the sample size n.