Question

Problem 1: Drawing from an Urn (no posted data set)

We will be comparing empirical probabilities (relative frequencies based on an observation of a real-life process) to theoretical probabilities (long-run relative frequency). We will use StatCrunch to simulate this process of drawing colored balls from an urn without replacement. Imagine this urn has 50 total balls, 18 of which are red and 32 of which are green. You draw 6 balls from the urn and we are interested in the number of red balls that are drawn.

1. The probability distribution table for the random variable representing the number of red balls drawn from the urn is given in the table below. Using one to two complete sentences, verify that the two features of a discrete probability distribution are valid.

Outcome	0	1	2	3	4	5	6
Probability	0.0570	0.2281	0.3462	0.2547	0.0955	0.0173	0.0012

2. Use the table in part (a) to calculate the probability that at least 3 red balls are drawn. Show any necessary work.

3. Is the probability calculated in part (b) a theoretical or empirical probability? Justify your answer in a sentence.

4. We will use StatCrunch to simulate drawing six balls from the urn. Start by opening a blank StatCrunch page by clicking the Open StatCrunch button on the home page of StatCrunch. Conduct the simulation by following the steps below:

          Step 1: Select Applets à Simulation à Urn sampling

Step 2: Under “Fill urn with balls:” in the “Number” column, enter 18 for Type 1 (with Color red) and 32 for Type 2 (with Color green).

Step 3: Under “Sampling:”, enter 6 for the “Number of balls to draw.” Make sure you do not select the “Draw with replacement” box.

Step 4: Under “Tally type 1 balls in sample:”, change the options next to “Number” to the equivalent of at least 3.

                Step 5: Click Compute!

Step 6: Click the “1000 runs” button. This will simulate drawing 6 balls from the urn 1,000 times.

Copy the result into your document using Options à Copy.

5. The value under “Proportion” is a probability. Is this a theoretical or empirical probability? Justify your answer and compare it to the probability in part (b) in two to three sentences.

6. Now clear your results by clicking the reset button at the top of the applet. Next change your “# of red balls” to equals three by clicking the drop down arrow in the box next to “# of red balls” and selecting the equal sign (=). Then, click the Convergence button. A new graphic should be displayed that has the probability on the y-axis and number of runs on the x-axis. Click the “5 runs” button 10 times to simulate 50 selections of 6 balls being drawn. Copy and paste your entire applet image into your solutions.

7. In one sentence, put the three values at the top of your image (Count, Total, and Proportion) in the context of the question.

8. Click reset. Now, click the “1000 runs” button 5 times to simulate 5000 selections of 6 balls. Copy and paste your entire applet image into your solutions.

1. Compare the proportions found in part (f) and (h) to the probability found above in the probability distribution where P(X = 3). Which of your proportions is closer to the given probability? Answer these in two to four sentences.

10. Which law of probability is being displayed with these images? Answer in one sentence.

Answer 1

a) Since all the values across the Probability row are positive and

So the given distribution is a a discrete probability distribution

b) Required probability is

c) As the above probability is calculated from the discrete probability distribution given in part a) so its a theoretical probability

d) No Access to StatCrunch