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MATH1127: Introduction to Statistics (60022) Lesson 7.7 Using the Central Limit Theorem (Combination)
Unit 3 Chapter 7: Lesson 7.7 Assignment
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A population of values has a normal distribution with $\mu=221.4$ and $\sigma=29.7$.
a. Find the probability that a single randomly selected value is between 225.5 and 226.3 . Round your answer to four decimal places.
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Solution
Step 1 ：We are given a population of values with a normal distribution, where the mean (\(\mu\)) is 221.4 and the standard deviation (\(\sigma\)) is 29.7.
Step 2 ：We are asked to find the probability that a single randomly selected value is between 225.5 and 226.3.
Step 3 ：To solve this, we need to standardize the values 225.5 and 226.3 using the formula: \[Z = \frac{X - \mu}{\sigma}\] where X is the value, \(\mu\) is the mean and \(\sigma\) is the standard deviation.
Step 4 ：Substituting the given values into the formula, we get: \[Z_1 = \frac{225.5 - 221.4}{29.7} = 0.138\] and \[Z_2 = \frac{226.3 - 221.4}{29.7} = 0.165\]
Step 5 ：We can then use the standard normal distribution (Z-distribution) to find the probabilities. The probability for \(Z_1\) is 0.555 and for \(Z_2\) is 0.566.
Step 6 ：The probability that a single randomly selected value is between 225.5 and 226.3 is the difference between these two probabilities, which is \(0.566 - 0.555 = 0.011\).
Step 7 ：Rounding to four decimal places, the final answer is \(\boxed{0.0106}\).
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Solution
Step 1 ：We are given a population of values with a normal distribution, where the mean (\(\mu\)) is 221.4 and the standard deviation (\(\sigma\)) is 29.7.
Step 2 ：We are asked to find the probability that a single randomly selected value is between 225.5 and 226.3.
Step 3 ：To solve this, we need to standardize the values 225.5 and 226.3 using the formula: \[Z = \frac{X - \mu}{\sigma}\] where X is the value, \(\mu\) is the mean and \(\sigma\) is the standard deviation.
Step 4 ：Substituting the given values into the formula, we get: \[Z_1 = \frac{225.5 - 221.4}{29.7} = 0.138\] and \[Z_2 = \frac{226.3 - 221.4}{29.7} = 0.165\]
Step 5 ：We can then use the standard normal distribution (Z-distribution) to find the probabilities. The probability for \(Z_1\) is 0.555 and for \(Z_2\) is 0.566.
Step 6 ：The probability that a single randomly selected value is between 225.5 and 226.3 is the difference between these two probabilities, which is \(0.566 - 0.555 = 0.011\).
Step 7 ：Rounding to four decimal places, the final answer is \(\boxed{0.0106}\).
