PROBABILITY AND STATISTICS
PROBABILITY AND STATISTICS
OBJECTIVES The lab is provided to introduce the student to the concepts of probability and sampling of a population of data. Probability is often used in science to predict outcomes of events.
BACKGROUND STATISTICS is the study of the average value (or other indicator values) of a large population of data. Because of natural range in variability, a few measurements of a natural phenomenon may not represent the average value of all possible occurrences. (For example, if you measure one chicken, it probably doesn’t represent the average of all chickens.) IN STATISTICS, MANY MEASUREMENTS OF THE OBJECTS OR EVENTS ARE REQUIRED FOR AN EXPERIMENT TO BE CONSIDERED VALID. PROBABILITY is the likelihood that an outcome may occur. What is the probability of rain for tomorrow? Well, probability is determined in two primary ways: 1. If there is an equal chance for any one of several outcomes to occur, the probability of just one occurring is 1 divided by the total number of possibilities. For example, there are 6 sides on a die from a dice game. The die is a cube with 6 equal sides, so on a toss there is an equal chance for any side to turn up. That is a probability of a “2″ turning up 1 out of 6 times, or .167 times. Another way to state this is that 16.7% of the time a “2″ will turn up. For events for which there are not equal chances for each outcome, a calculation is either very difficult or impossible. In this case, probability is determined by staging very many trials. The event is observed countless times, and the frequency of occurrence of the desired outcome is used as the predictor of the probability. The number of successful outcomes (e.g., a “2″ turning up) is divided by the total number of trials (total number of throws) to get the frequency of occurrence (e.g., out of 300 throws the “2″ came up 50 times for a frequency of .167 or 16.7%). This frequency can be used as a predicting probability as long as there is a sufficient number of trials. Experience has shown that the more trials used in calculating frequency of occurrence, the more reliable the value is as a probability estimate. The weather man has over 50 years of records and finds the frequency of occurrence of rain under different conditions. He uses this to help predict weather when those conditions arise again.
EQUIPMENT You may choose to flip a coin or toss a die (cast lots or draw straws) according to what you have on hand. A calculator will be convenient but is not necessary.
PROCEDURES 1. First, use 10 trials to build a sample of outcomes. Record how many times your chosen number or a head came up. Continue flipping in groups of 10, calculating your cumulative results as you go (e.g., results after 10, 20, 30, 40, etc.), until you have run 400 trials. 2. Calculate the frequency of occurrence for each experiment by dividing the number of times the desired outcome occurred by the number of tosses (e.g., number of heads out of 10 flips, then number of heads out of 20 flips, then number of heads out of 30 flips, etc.). Calculate the theoretical value of your probability by dividing the number of possible outcomes into 1. Compare your frequency values from 10 trials, 50 trials, 200 trials, and 400 trials to the theoretical value. A precise way to compare your outcome to the theoretical value is by the Experimental Error Formula. Use the following formula to express your variation as a percent. Variation (or Experimental Error)
Theoretical Value – Experimental Value x 100 Theoretical Value
For example: I got .525 for my frequency of heads.
.5 – .525 x 100 = .05 x 100 = 5% .5
RESULTS AND CONCLUSIONS Report your results in a neat table, and include your calculations. Make a graph of the number of trials versus the frequency of occurrence for 10, 50, 100, 200, 300, and 400. Does your graph tell you the point where you have a large enough number of trials? Summarize your results and conclusions.
APPLICATION If two opinion polls were taken regarding the next presidential election, should you believe the one involving 71,502 people from the suburbs or the one in which 215 people from the city were polled? Why?
laboratory report should contain the following sections: (1) Hypothesis, (2) Procedures,
(3) Observations and Results, and (4) Conclusions. Make certain you include all four headings with at least a short paragraph for each. In addition, tables, graphs, and answers to questions may be necessary in the latter two sections.
Scientific research should contain a preliminary statement of the expected outcome of the experiment. This can include predictions of the specific experiment or the general anticipated result. If you are merely doing an observation and have no idea of the outcome, you cannot make an actual hypothesis. Instead, make a short statement of the purpose of the observation. However, if you have preconceived ideas of the outcome, include them in this section, and then see how they compare to the results.
Even though you are told what to do, write a paragraph of the specific steps you actually took in doing the experiment or observation. Because you are coming up with your own equipment, your procedures will be of particular interest.
OBSERVATIONS AND RESULTS
This is where you should make a detailed statement of the outcome of your experiment. Record all your pertinent observations in a clear, readable form. Arrange your data in tables (such as measurements and calculations you make). Answer any questions asked in this Study Guide, marking these clearly so that they can be easily found.
Your conclusions should include a comparison between the outcome of the experiment and your initial predictions made in the hypothesis. In cases where you are attempting to recreate a physical constant, compare your number to the accepted value, using the formula for experimental error:
Experimental Error Equation
If you find a large difference in your results from the expected value or if your anticipated observations are not the same as your actual observations, try to identify possible sources of error or reasons for the difference in the hypothesis and results