Fundamentals of Statistics Project

This project will verify the randomness of the calculator’s random number generator. The calculator comes with a feature that lets it generate a random number. Your goal is verify if the calculator will generate numbers randomly when asked to generate a number from 1 through 5. If it is truly random, it should follow a uniform distribution (meaning all numbers have equal chance of showing). We will check the number of times the calculator generates the number 3. To do this we will:

1. Generate a sample from the calculator.
   1. Hit [prb]
   2. Go to RAND
   3. Pick option 2: randint(
   4. Enter in [1] [2nd] [ . ] 5
   5. Press enter to generate data.
   6. Record which attempt you are on and the value the calculator generated.
      i.e.: 1(first attempt): 3(what calculator generated)
   7. Keep pressing enter to generate more data. Record 100 values this way from the calculator.
2. Analyzing Binomial Distribution
   1. Calculate the probability of getting a 3 in one attempt. Recall what numbers the calculator can generate.
   2. Calculate the expected value (mean of binomial) for the numbers 3’s we should expect to get:
      1. For a sample size of 50
      2. For a sample size of 100
   3. Calculate the standard deviation
      1. For sample size 50
      2. For sample size 100
   4. Calculate the interval of normal values (for getting a 3)
      1. For sample size 50
      2. For sample size 100
3. Confidence Intervals (based on your sample)
   Create confidence intervals based on the different sample sizes and confidence levels based on the proportion of times you got 3’s in your sample.
   1. Confidence interval based on sample size 50, Confidence level 90%
   2. Confidence interval based on sample size 50, Confidence level 99%
   3. Confidence interval based on sample size 100, Confidence level 90%
   4. Confidence interval based on sample size 100, Confidence level 99%
4. Making conclusions based on our analysis.
   1. Compare the two margin of errors for sample size of 50 with the two margin of errors for sample size 100.
      Restate the 2 margins of errors for sample size 50 and the two margins of errors for sample size 100. 
      1. What did you notice about the of margins of errors as the sample size went up? Are the margins of errors for sample size 50 bigger or are the margins of errors for sample size 100 bigger?
      2. Which version is the better version? The ones margins of errors for sample size 50 or for sample size 100? Explain why it is better.
   2. Did the number of 3’s in your sample fall in the interval of normal values based on the binomial distribution? Restate the IoNV and the number of 3’s you had. Followed by – YesNo
      1. For sample size 50
      2. For sample size 100
   3. Did each confidence interval do their job in containing the expected proportion (this is the probability of getting one 3 in one attempt that we calculated in part 1 of binomial distribution)?
      Restate each confidence interval along with what sample size and what confidence level was used. Followed by –  YesNo.

This project touches on topics in hypothesis testing. We know the expected value, and run trials to see if we get a number of 3’s that falls in the interval of normal values. We also make confidence intervals to see if we can correctly guess the population proportion using samples. If we get results that are not in the interval of normal values or we get confidence intervals that do not contain the expected proportion, then we question if the expected values are actually the true values.

For example, a game claims you will win 70% of the time if you play. If I play a game 1000 times and I expect to win 700 times, but I only win 100 times, I might question the claim that I will win 70% of the time. Maybe the game was lying to me.

Based on your results in the study, if you got a normal amounts of 3’s and the proportion of getting a 3 was in all of your confidence intervals, then we can safely assume the calculator is using a random number generator. If not, we may have evidence to suspect that the calculator is not using a random number generator.

Your data and calculations should be neat and organized. And your conclusion must reference your data.

Grade distribution:

10% – Generating your samples

80% – Making accurate analyzations

5% – Making a correct conclusion

5% – Neatness and organization of data and calculations; Overall quality of the project

For a total of 100%

The project is worth 5% of the total semester grade. You may ask me for help and work with others for help doing your calculations. But, you must have your own sample and work in the end.