Some important statistics

• Random samples are selected to elicit information about the unknown population parameters.
• Some important statistics:
  • sample mean
  • sample variance
• Definition 8.4:
  
• Say $p$ is a function of the observed values in the random sample.
• We would expect $p$ to vary somewhat from sample to sample.
• That is a value of a random variable $P$, called a statistic.

• Definition 8.5:
  
• The mean, median, and mode are the most commonly used statistics for measuring the central tendency.
• The computed value of $\bar{X}$ for a given sample is denoted by $\bar{x}$.
• Sample mean is not the same thing as the mean of a random variable but they are very closely related.
• Sample mode is the observation value that occurs the most number of times in a sample.
• Sample median is the middle value of a sample after sorting.

• Definition 8.6:
  
• The computed value of $S^2$ for a given sample is denoted by $s^2$.
• Again this is very related to the standard deviation of a random variable but is not the same thing.

• Example 8.1: A comparison of coffee prices at 4 randomly selected grocery stores in San Diego showed increases from the previous month of 12, 15, 17, and 20 cents for a 1-pound bag.
• Find the variance of this random sample of price increases.
• Solution:
  $$\bar{x}=\frac{12+15+17+20}{4}=16$$
  $$s^2=\frac{\sum_{i=1}^4(x_i-16)^2}{4-1}=\frac{(12-16)^2+(15-16)^2+(17-16)^2+(20-16)^2}{3}=\frac{34}{3}$$

• Theorem 8.1
  

• Definition 8.7:
  
• Example 8.2: Find the variance of the data 3, 4, 5, 6, 6, and 7, representing the number of trout caught by a random sample of 6 fishermen.
• Solution:
  $$\sum_{i=1}^6 x_i^2=171$$
  $$\sum_{i=1}^6 x_i=31$$
  $$\sigma^2=\frac{6*171 -31^2}{6*5}=\frac{13}{6}$$