In a large class, the professor has each person toss a coin several times and calculate the proportion of his or her tosses that were heads.The students then report their results, and the professor plots a histogram of these several proportions.If the students toss the coin 80 times each, about 95% should have proportions between what two numbers?
Use the 68-95-99.7 Rule to provide the appropriate response.
A)0.025 and 0.975
B)0.39 and 0.62
C)0.1118 and 0.1677
D)0.4875 and 0.5125
E)0.2375 and 0.7375

Question

Quizplus · Accepted Answer

The 68-95-99.7 Rule, also known as the Empirical Rule, applies to normal distributions and states that about 95% of the data falls within two standard deviations of the mean. For a fair coin toss, the mean proportion of heads is 0.5. The standard deviation for the proportion of heads in 80 tosses can be calculated using the formula for the standard deviation of a binomial distribution, which is sqrt[p(1-p)/n], where p is the probability of success (0.5 for a fair coin), and n is the number of trials (80). This gives sqrt[0.5*0.5/80] = sqrt[0.25/80] = sqrt[0.003125] ≈ 0.0559. Therefore, two standard deviations is 2*0.0559 = 0.1118. Adding and subtracting this from the mean (0.5) gives us the range 0.5 ± 0.1118, which is approximately 0.3882 to 0.6118, closest to the range given in option B (0.39 to 0.62).

In a Large Class, the Professor Has Each Person Toss