In a recent year, the scores for the reading portion of a test were normally distributed, with a mean of 23.2 and a standard deviation of 5.7. Complete parts (a) through (d) below.
(a) Find the probability that a randomly selected high school student who took the reading portion of the test has a score that is less than 21
The probability of a student scoring less than 21 is (Round to four decimal places as needed.)
Answer
Final Answer: The probability of a student scoring less than 21 is approximately \(\boxed{0.3498}\).
Step 1 ：Given that the scores for the reading portion of a test were normally distributed, with a mean of 23.2 and a standard deviation of 5.7.
Step 2 ：We are asked to find the probability that a randomly selected high school student who took the reading portion of the test has a score that is less than 21.
Step 3 ：This is a problem of finding the probability of a value in a normal distribution. The formula for finding the z-score is: \(z = \frac{X - \mu}{\sigma}\) where X is the value we're interested in (21 in this case), \(\mu\) is the mean (23.2), and \(\sigma\) is the standard deviation (5.7).
Step 4 ：Substituting the given values into the formula, we get: \(z = \frac{21 - 23.2}{5.7} = -0.3859649122807016\)
Step 5 ：Once we have the z-score, we can use a z-table or a function like scipy's norm.cdf() to find the probability.
Step 6 ：The probability of a student scoring less than 21 is approximately 0.3498. This means that about 34.98% of students are expected to score less than 21.
Step 7 ：Final Answer: The probability of a student scoring less than 21 is approximately \(\boxed{0.3498}\).
