The random variable X is such that X ~ N(82, 126).
A value of X is chosen at random and rounded to the nearest whole number. Find the probability that this whole number is 84.
Solution
To find the probability that a value of X rounds to 84, we need to calculate the probability that X is between 83.5 and 84.5.
Standardize the values using the formula for the standard normal distribution:
\(Z = \frac{X - \mu}{\sigma}\)
where \(\mu = 82\) and \(\sigma = \sqrt{126}\).
Calculate the standardized values:
\(Z_1 = \frac{84.5 - 82}{\sqrt{126}} = 0.2227\)
\(Z_2 = \frac{83.5 - 82}{\sqrt{126}} = 0.1336\)
Find the probabilities using the standard normal distribution table:
\(\Phi(0.2227) = 0.5883\)
\(\Phi(0.1336) = 0.5533\)
Subtract the probabilities to find the probability that X rounds to 84:
\(P(83.5 < X < 84.5) = \Phi(0.2227) - \Phi(0.1336) = 0.5883 - 0.5533 = 0.0350\)
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