Suppose you roll a die 10 times and record the proportion of sixes. Suppose you then conduct a simulation of this experiment, first 100 times, then 1000 times, and draw one histogram of the proportion of sixes found after 100 simulations and a second histogram of the proportions of sixes found after 1000 simulations.
Which of the following statements are true regarding the histograms of the results from the two simulations?
I.The histograms from both simulations will be skewed left since a fair die does not exist in nature.
II.The histograms from both simulations will be mound-shaped and symmetric.
III. The histogram from the experiment that has 1000 simulations will tend to be more mound-shaped and symmetric than the histogram from the experiment that has 100 simulations.
a.I only
b. I and II
c. II and III
d. III only
e. None of the statements is true.
Im thinking either c or e because the probability of getting a 6 is 1/6 for every simulation.
I would go with d, however, it depends on the words "more" mound-shapped and symettric
To determine which statements are true regarding the histograms of the results from the two simulations, let's break down each statement and analyze it:
I. The histograms from both simulations will be skewed left since a fair die does not exist in nature.
Explanation: Skewness is a measure of the asymmetry of a distribution. In this case, since the die is fair and unbiased, the probabilities of rolling any number, including six, should be equal. Therefore, there is no reason for the distribution to be skewed left specifically. This statement is false.
II. The histograms from both simulations will be mound-shaped and symmetric.
Explanation: When rolling a fair die, all possible outcomes (numbers 1 to 6) have equal probabilities. In the long run, the proportion of sixes should converge to the expected probability of 1/6. As the number of simulations increases, the distribution of the proportions should tend to a mound-shaped, symmetric distribution around the expected probability. This statement is true.
III. The histogram from the experiment that has 1000 simulations will tend to be more mound-shaped and symmetric than the histogram from the experiment that has 100 simulations.
Explanation: As mentioned earlier, increasing the number of simulations helps to approach the expected probability more accurately. With 1000 simulations, the sample size is larger compared to 100 simulations, allowing for a better estimate of the true proportion of sixes. Therefore, the distribution will likely be more mound-shaped and symmetric. This statement is true.
Based on the analysis, the correct answer is c. II and III