A multiple-choice test consists of 29 questions with possible answers of a, b, c, d. Estimate the probability that with random guessing, the number of correct answers is at least 14.

z-value (Use continuity correction)=
Probability=

To find the probability that with random guessing, the number of correct answers is at least 14, we can use the normal approximation to the binomial distribution.

First, let's calculate the mean and standard deviation of the number of correct answers.

Mean (μ) = n * p
Here, n is the number of questions (29) and p is the probability of getting a correct answer by random guessing (1/4, as there are 4 possible choices for each question).

μ = 29 * (1/4) = 7.25

Standard Deviation (σ) = sqrt(n * p * (1-p))
Using the same values of n and p:

σ = sqrt(29 * (1/4) * (3/4)) ≈ 2.388

Next, we need to calculate the z-value using the continuity correction.

Z-value (Z) = (x - μ - 0.5) / σ
Here, x is the minimum number of correct answers we want (14) and 0.5 is the continuity correction factor.

Z = (14 - 7.25 - 0.5) / 2.388 ≈ 2.765

Now that we have the z-value, we can find the probability using a standard normal distribution table or calculator. Since we want the probability that the number of correct answers is at least 14, we need to find the area to the right of the z-value.

Probability = 1 - P(Z ≤ 2.765)

You can use a standard normal distribution table or calculator to find this probability. For example, using a calculator, you can enter the z-value (2.765) and find the corresponding probability (0.0029).

So, the estimated probability that with random guessing, the number of correct answers is at least 14 is approximately 0.0029.