Given that the population mean is 188 points and the variance is 361, we can find the standard deviation as follows:
Standard deviation = square root of variance = square root of 361 = 19
The standard error of the mean is given by the formula:
Standard error of the mean = standard deviation / square root of sample size
Standard error of the mean = 19 / square root of 73 = 2.2295 (rounded to four decimal places)
To find the probability that the mean of the sample would differ from the population mean by less than 3.7 points, we need to find the z-score using the formula:
z = (x - μ) / (σ / sqrt(n))
where x is the sample mean, μ is the population mean, σ is the standard deviation, and n is the sample size.
Substituting the given values, we get:
z = (x - 188) / (19 / sqrt(73))
We want to find the probability that the difference between the sample mean and the population mean is less than 3.7 points, which means we want to find the probability that:
|x - 188| < 3.7
This is equivalent to finding the probability that:
-3.7 < x - 188 < 3.7
Simplifying this inequality, we get:
184.3 < x < 191.7
Substituting the lower and upper limits of x in the z-score formula, we get:
z1 = (184.3 - 188) / (19 / sqrt(73)) = -1.7376
z2 = (191.7 - 188) / (19 / sqrt(73)) = 1.7376
Using a standard normal distribution table, we can find the area under the curve between z1 and z2 as follows:
Area = P(z1 < z < z2) = P(z < 1.7376) - P(z < -1.7376) = 0.9589 - 0.0411 = 0.9178 (rounded to four decimal places)
Therefore, the probability that the mean of the sample would differ from the population mean by less than 3.7 points if 73 exams are sampled is 0.9178 (rounded to four decimal places).