AGRI 2400 Lecture 14-15 - Samples and the Normal Distribution

Populations and Samples

• population mean, variance, and SD are population parameters

  • they are true values obtained by measuring every individual of a population

• when you take a sample from a pop and then calculate a mean, variance, or SD, these are only estimates of that population parameter

Standard Error of the Mean

• abbreviated as either SEM or SE

• symbol is ox(bar)

• calculated by taking the standard deviation of the population and dividing it by the square root of the sample size

• used to calc. the range within which a particular percentage of the sample means will occur

• since sample means are normally distributed with a mean of population mean

Z Statistic

• as the difference between X(bar) and mew increases, the absolute value of the Z statistic increases

• when Z is less than -1.96 or greater than +1.96, the probability of observing the difference between the sample mean and the known population mean, or an even larger difference, is less than 5%

Z Statistic and Hypothesis Testing

• based on our Z statistic falling outside the -1.96 to 1.96 range, what would we conclude?

  • the sample mean is significantly different than the pop mean it is being compared to, meaning that the mean yields of the new variety and standard variety do not appear to be equal

What About Unknown Pop Parameters?

• in most cases, mew and o (the population parameters) are unknown

• but we do have estimates of the population parameters (X(bar) and s)

  • these parameters can also be used to estimate the SEM

SEM Using Sample SD (s)

• this estimate of the standard error of the mean, based on sample parameters, can also be used to predict the range around any hypothetical value of mew within each 95% of the means of samples of size n taken from that population will occur

  • this is called the 95% confidence interval

Law of Large Numbers

• our confidence that our estimate of pop. mean (mew is close to the real pop. mean increases with sample size

Student’s t Distribution

• the student’s t distribution accounts for the effect of sample size on the confidence interval, and is used for hypothesis testing instead of the Z distribution when population parameters are unknown

Central Limit Theorem

• even when a population is not normally distributed, if you take repeated samples of 25 or more, the distribution of sample means will have an approx. normal distribution

  • for pops that are approx. normal to begin with, this property may hold for sample sizes as low as 5

• this property of samples makes it possible to use statistical analyses that assume a normal distribution, when sampling pops that are not normally distributed, if your sample size is adequate

Median

• the value of a set of ordered observations that has an equal number of observations above and below it

  • for an odd # of obs, it is the central most observation

  • for an even # of obs, it is the midpoint between the 2 central obs

Mode

• the observation that occurs most frequently in the sample

Range

• smallest and largest value from a sample

Quantiles

• values which divide a distribution such that there is a given proportion of observations below the quantile

Quartiles

• are the most commonly calculated quantiles

  • the quartiles divide the distribution into 4 equal parts

    • Q1 = 25th perctile (lower)

    • Q2 = median (central)

    • Q3 = 75th percentile (upper)