Stats Chapter 7

Sample Mean and Population Mean

• The sample mean is an estimator of the population mean. It serves as a single point estimate.

Sample Proportion and Population Proportion

• The sample proportion acts as an estimator for the population proportion.

Sample Variance and Population Variance

• Sample variance and sample standard deviation are estimates of population variance and population standard deviation, respectively.

Point Estimators

• These estimates (sample mean, sample proportion, etc.) are referred to as point estimators.
• Point estimators provide single numerical values to estimate population parameters.

Properties of Point Estimates

• Point estimates are expected to have properties concerning bias, consistency, and efficiency.
• Due to the imprecision of point estimates, interval estimates are often employed, leading to the concept of confidence intervals.

Interval Estimates

• An interval estimate provides a range of values rather than a single value to estimate a parameter (e.g., sample mean less than 20).
• Example of interval: A sample mean might be said to lie between 15 and 25.

Best Point Estimators

• For different parameters, the best point estimators are defined as:
• Population mean: Use the sample mean.
• Population proportion: Use the sample proportion.
• Population variance: Use the sample variance.

Finding Interval Estimates

• There are two cases for finding interval estimates for the population mean:

1. When the population standard deviation is known.
2. When the population standard deviation is unknown.

Known Population Standard Deviation

• The formula for confidence interval for the population mean when the population standard deviation \sigma is known is given by:
  \text{CI} = \bar{x} \pm z_{\alpha/2} \left( \frac{\sigma}{\sqrt{n}} \right)
• Where \bar{x} is the sample mean, z_{\alpha/2} is the critical value from the normal distribution, \sigma is the population standard deviation, and n is the sample size.

Confidence Levels

• For a 90% confidence interval, z_{\alpha/2} is approximately 1.645.
• For a 95% confidence interval, z_{\alpha/2} is approximately 1.96.
• For a 99% confidence interval, z_{\alpha/2} is approximately 2.576.

Margin of Error

• The margin of error E is computed as:
  E = z_{\alpha/2} \left( \frac{\sigma}{\sqrt{n}} \right)
• To determine sample size n considering the margin of error, rearranging gives:
  n = \left( \frac{z_{\alpha/2} \sigma}{E} \right)^2

Example Calculation

• Consider determining sample size with a given margin of error for different confidence levels. An example with \sigma = 1.7,\ z_{\alpha/2} = 1.65, and E = 0.5 leads to calculations to find the sample size.

Transition to Unknown Population Standard Deviation

• When the population standard deviation is unknown, we shift from using the standard normal distribution to the t-distribution.
• The t-distribution is similar to the normal distribution but varies based on degrees of freedom (df = n - 1).

Finding T-Critical Values

• The t-value can be found using a t-table based on degrees of freedom and confidence level.
• The formula for confidence interval changes to:
  \text{CI} = \bar{x} \pm t_{\alpha/2} \left( \frac{s}{\sqrt{n}} \right)
• Where s is the sample standard deviation.

Practical Example

• Example: A nutritionist gathered a sample of 60 adults to assess candy consumption per year, finding a sample mean and applying formulas to calculate the confidence interval for different confidence levels.