Chapters 6 & 7 Project

Get ready to apply your knowledge! Here's the project for Chapters 6 and 7. This project will help you solidify your understanding of key statistical concepts like normal distributions, standard deviations, and sampling techniques. Let's tackle this project and reinforce your learning!

Part 1: Family Heights and the Normal Curve

In this part, you'll explore how your family's heights compare to the normal distribution. Specifically, you'll:

  1. Determine how many standard deviations each family member's height is above or below the mean and their corresponding percentile.
  2. Investigate if there's a relationship between parents' and children's heights.
  3. Using the appropriate gender-specific chart, determine the heights within one standard deviation of the mean, the record holder's deviation above the mean, and the smallest person's deviation below the mean.

Part 2: The Chesapeake and Ohio Freight Study

This section delves into a real-world application of sampling. You'll analyze a freight study by the Chesapeake and Ohio Railroad Company (C & O), where a sample of 2072 waybills was used to estimate the total revenue due. You'll address questions such as:

  1. What percentage of waybills constituted the sample?
  2. What percentage error was made in estimating the total revenue using the sample?
  3. Considering the cost of a census versus the sample estimate, was sampling preferable? Why?
  4. Could the $83 error in the study have been in C & O's favor?

Chapter 8 Class Notes: Estimation and Confidence Intervals

Let's move on to Chapter 8, focusing on point estimates and confidence intervals. These concepts are crucial for making inferences about populations based on sample data.

Point Estimate

A point estimate is a single value that best approximates a population parameter. For example, the sample mean, denoted as $\bar{x}$, is a point estimate of the population mean, denoted as $\mu$.

Example: Prices of New Mobile Homes

Consider the prices of 36 randomly selected new mobile homes. The data (in thousands of dollars) is used to estimate the population mean price, $\mu$, of all new mobile homes.

Given the sample data, the sample mean is calculated as:

$$ \bar{x} = \frac{\Sigma x_i}{n} = \frac{2278}{36} = 63.28 $$

Therefore, the point estimate for the population mean price is $63,280.

Confidence Intervals

A confidence interval (CI) provides a range of values within which the population parameter is likely to fall. It's constructed using a point estimate and a margin of error.

Key terms:

  • Confidence Level: The probability that the confidence interval contains the population parameter. For example, a 95% confidence level means that if we were to take 100 different samples and compute a confidence interval for each sample, we would expect approximately 95 of the intervals to contain the true population mean.
  • Margin of Error: Represents the amount by which the point estimate might differ from the population parameter.

The general form of a confidence interval is:

$$ Point\ Estimate \pm Margin\ of\ Error $$

One-Mean z-Interval Procedure

To find a confidence interval for a population mean, $\mu$, when the population standard deviation, $\sigma$, is known, we use the z-interval procedure.

The formula for the confidence interval is:

$$ \bar{x} - z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}} \text{ to } \bar{x} + z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}} $$

Where:

  • $\bar{x}$ is the sample mean.
  • $z_{\alpha/2}$ is the z-score corresponding to the desired confidence level.
  • $\sigma$ is the population standard deviation.
  • $n$ is the sample size.

Formula for Margin of Error

$$ E = z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}} $$

Sample Size for Estimating $\mu$

To determine the necessary sample size for a specific margin of error, $E$, the formula is:

$$ n = \left( \frac{z_{\alpha/2} \cdot \sigma}{E} \right)^2 $$

Remember to always round up to the nearest whole number.

Keep practicing and exploring these concepts – you've got this!