Probability and Statistics Objectives
- Obtain organizational procedures for analyzing data.
- Recognize data that can be classified as empirical.
- Conduct a random sample experiment by describing and selecting a sample from a population.
- State the definition for simple and compound events; list examples of each kind of event.
- Make reasonable conjectures from purely chance phenomena.
- Express the probability of an event as a fraction.
- State the definition for the probability of an event.
- Compute the probability of an event, P(E), in a finite sample by summing the probabilities of the sample points in the event.
- Acquire an efficient method for counting arrangements of objects.
- Identify permutations and combinations as counting events in a sample.
- Use the permutation and combination theorems in problem solving.
- Describe relationships between two or more events.
- Classify two or more events as one or more of the following:
- mutually exclusive
- State the definition for the conditional probability of two events A and B as: P(B/A) = P(AB)/P(A) or P(A/B) = P(AB)/P(B).
- Prove that two events are independent if either P(A/B) = P(A) or P(B/A) = P(B).
- Predict the outcome of a combination of events given the results of at least one of those events.
- Compute the conditional probability of two events that are either dependent or independent.
- Use the following laws of probability in calculating the probability of a compound event:
- additive: P(A + B) = P(A) + P(B) - P(AB)
- multiplicative: P(AB) = P(A)P(B/A) or P(B)P(A/B)
- Mathematics of Apportionment
- Express sums in a concise form.
- Use correctly the symbol Σ
- Use the following theorems as they apply to a given sum:
|Σ||cxi = n||Σ||xi|
|Σ||xi + yi - zi = ||Σ||xi + ||Σ||yi - ||Σ||zi|
- Report the information contained in data.
- Prepare and use histograms.
- Use the terms “population” and “sample” correctly.
- Find for a given experiment the following measures of central tendency:
- mean [ (sample), μ (population)]
- Find for a given experiment the following measures of variability:
- variance [ s2 (sample variance with (n-1) degrees of freedom), σ2 (population)]
- standard deviation [ σ (population), s (sample)]
- Compute the mean and the sample variance, when given a population sample.
- Use the sample mean and the sample variance to estimate the population mean and population variance.
- State and apply the empirical rule as the rule relates to Chebyshev's theorem.
- Probability Distribution
- Apply the past success of an event to future trials.
- State the four requirements for a binomial experiment.
- Derive the binomial probability distribution
- Compute the binomial probability distribution for a given experiment.
- Compute the mean and standard deviation for the binomial variable in a given binomial experiment.
- Minimize the chance of error in accepting or rejecting a given hypothesis.
- Construct graphically the operating characteristic curve for a given sampling plan.
- Define for a given experiment the following:
- null hypothesis (H0)
- acceptance region
- rejection region
- Type I errors (α) and Type II errors (β)
- Statistical Inference
- Draw useful inferences from samples applied to the normal distribution.
- State the central limit theorem and apply the theorem to a given experiment.
- Construct the normal probability distribution curve and interpret its tabulated values as areas lying within a specified number of standard deviations of the mean.
- Approximate the binomial distribution by use of the normal distribution and compute the following for a given experiment:
- values of z (the standardized normal variable)
- probabilities for x (the random variable)
- probability of a Type I error and location of the rejection region
- probability of a Type II error and location of the acceptance region
- Distinguish between a confidence interval and a confidence coefficient.
- Demonstrate successful sampling procedures to evaluate a hypothesis.
- State the hypothesis to be tested.
- Select the sample.
- State the alternative hypothesis.
- Evaluate Type I or Type II errors.
- Enrichment Objectives
- Demonstrate successful sampling procedures to evaluate a hypothesis involving two populations by completing each of the following:
- Test a hypothesis concerning µ (the mean) of two populations by use of the student’s distribution.
- Compute and make inferences from small samples concerning the difference between two means.
- Identify the Chi-square probability distribution and the F-distribution and interpret tabulated values for each kind of distribution.
- Draw inferences from the variance of two given populations by using
the Chi-square test and the F-test.
- Linear Regression and Correlation
- Use a straight line as a predictor for the outcome of an experiment.
- Construct a scatter plot for a given experiment.
- Obtain a prediction equation for a given experiment by use of the scatter plot and a line of “best fit.”
- Use the least squares method to find the “best” fitting line for a given set of points.
- Compute and analyze the variance of the random error of the deviation from the regression line by using the data from a given experiment.
- Use the regression equation y = β0 + β1x to:
- Test H0: β1 = 0 against H : β1 ≠ 0.
- Estimate the slope β1.
- Compute the expected value of y, given x[E(y/x)].
- Compute the coefficient of correlation between y and x for a given line.
- Define and use a prediction equation for a given experiment.