"Birth Weights Suppose birth weights of babies are approximately normal with a mean of 3570 grams and a standard deviation of 495 grams. In a random sample of 15 mothers who were smokers during pregnancy, the mean birth weight of the babies was 3481.6 grams.
b. Simulate obtaining 2000 simple random samples for size n=15 babies from a population that is normally distributed with mean 3570 and standard deviation 495. For each sample, determine the sample mean. Draw a histogram of the 2000 sample means. Explain what each sample mean represents."

Verified step by step guidance

1. Understand the problem context: We have a population of birth weights that is normally distributed with mean \( \mu = 3570 \) grams and standard deviation \( \sigma = 495 \) grams. We want to simulate the process of taking samples of size \( n = 15 \) from this population and calculate the sample means.
2. Simulate 2000 simple random samples, each of size 15, from the normal distribution \( N(3570, 495^2) \). This means for each sample, randomly generate 15 birth weights using the given mean and standard deviation.
3. For each of the 2000 samples, calculate the sample mean by summing the 15 values and dividing by 15. This will give you 2000 sample means.
4. Create a histogram of these 2000 sample means. The histogram will show the distribution of the sample means, which should approximate a normal distribution due to the Central Limit Theorem.
5. Interpretation: Each sample mean represents the average birth weight of a group of 15 babies randomly selected from the population. The histogram shows the variability of these averages across many samples, illustrating the sampling distribution of the sample mean.

Key Concepts

Sampling Distribution of the Sample Mean

The sampling distribution of the sample mean is the probability distribution of all possible sample means of a given size drawn from a population. It describes how the sample mean varies from sample to sample and tends to be approximately normal if the population is normal or the sample size is large, regardless of the population distribution.

Simulation of Random Samples

Simulation involves generating many random samples from a specified population distribution to approximate the behavior of a statistic, such as the sample mean. By simulating 2000 samples of size 15, we can observe the variability and distribution of sample means without relying solely on theoretical formulas.

Histogram and Interpretation of Sample Means

A histogram visually displays the frequency distribution of the 2000 sample means, showing how often each range of means occurs. Each sample mean represents the average birth weight from one simulated sample of 15 babies, illustrating the variability and central tendency of sample means around the population mean.