To answer this question, we need to calculate the z-score, use the cumulative distribution function (CDF) to obtain the probability, and then estimate the number of students who will be rejected based on the calculated probability. Here is how we do it step by step:
Step 1: Calculation of the Z-Score
The z-score is calculated using the formula: z = (X - μ) / σ, where X is the data point (in this case, the required IQ), μ is the mean, and σ is the standard deviation.
Here, the required IQ (X) is 98, the mean IQ (μ) is 118, and the standard deviation (σ) is 13. Substituting these values into the formula, we get:
z = (98 - 118) / 13 which gives us a z-score of -1.54 (rounded to two decimal places).
Step 2: Probability Calculation
We then use the Cumulative Distribution Function (CDF) to calculate the probability. The CDF gives us the probability that a random variable is less than or equal to a certain value. Since we have a negative z-score, this gives us the probability of an applicant having an IQ less than 98.
By looking up the z-score of -1.54 in a standard normal distribution table or using a scientific calculator, we find that the probability is approximately 0.062.
Step 3: Estimation of the Number of Rejections
To estimate the number of students who will be rejected, we multiply the total number of students by the calculated probability.
Hence, 700 (total applicants) * 0.062 (probability) approximately equals 43.
Therefore, we estimate that about 43 applicants will be rejected based on their IQ score. This estimation is rounded to the nearest integer, since the number of people cannot be a decimal or fraction.