a quiz consists of 30 true or false questions. if the student guesses on each question, what is the standard deviation of the number of correct answers
The Range Rule
The range rule tells us that the standard deviation of a sample is approximately equal to one fourth of the range of the data. In other words s = (Maximum – Minimum)/4. This is a very straightforward formula to use, and should only be used as a very rough estimate of the standard deviation.
15
To calculate the standard deviation of the number of correct answers, we need to know the probability of guessing correctly and the probability of guessing incorrectly on a true or false question.
In this case, since the student is guessing, the probability of guessing each question correctly is 0.5, and the probability of guessing incorrectly is also 0.5.
Now, let's use the formula for the standard deviation of a binomial distribution, which is given by the square root of n * p * (1 - p), where n is the number of trials and p is the probability of success.
In this case, the number of trials is 30 (the number of questions), and the probability of success is 0.5 (the probability of guessing each question correctly).
Standard deviation (σ) = sqrt(n * p * (1 - p))
= sqrt(30 * 0.5 * (1 - 0.5))
= sqrt(30 * 0.5 * 0.5)
= sqrt(7.5)
≈ 2.74
Therefore, the standard deviation of the number of correct answers in a quiz of 30 true or false questions, where the student guesses on each question, is approximately 2.74.
To find the standard deviation of the number of correct answers, we need to calculate the variance first. The variance measures the variability or dispersion of a set of data points.
The expected value for each question is 0.5 since the student has a 50% chance of guessing correctly on each true or false question. Therefore, the expected value for the number of correct answers in this case is:
Expected value = (number of questions) * (probability of guessing correctly) = 30 * 0.5 = 15
Next, we need to calculate the variance using the formula:
Variance = (probability of outcome 1 * (outcome 1 - expected value)^2) + (probability of outcome 2 * (outcome 2 - expected value)^2) + ...
In this case, there are two possible outcomes: the student guesses correctly (1) or guesses incorrectly (0).
Probability of guessing correctly = 0.5
Probability of guessing incorrectly = 1 - 0.5 = 0.5
Variance = (0.5 * (1 - 15)^2) + (0.5 * (0 - 15)^2) = 0.5 * 14^2 + 0.5 * (-15)^2
Simplifying the calculations:
Variance = 0.5 * 196 + 0.5 * 225
Variance = 98 + 112.5
Variance = 210.5
Finally, we find the standard deviation by taking the square root of the variance:
Standard deviation = √210.5
Standard deviation ≈ 14.51
Therefore, the standard deviation of the number of correct answers is approximately 14.51.