In a recent year, the scores for the reading portion of a test were normally distributed, with a mear standard deviation of 6.4 . 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 20 . The probability of a student scoring less than 20 is 0.4251 . (Round to four decimal places as needed.) (b) Find the probability that a randomly selected high school student who took the reading portion of the test has a score that is between 14.0 and 28.4 . The probability of a student scoring between 14.0 and 28.4 is (Round to four decimal places as needed.)
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To tackle part (b), we first need the mean of the reading scores to calculate the probability that a randomly selected student scored between 14.0 and 28.4. Assuming the mean is known and can be added (let's denote it as \( \mu \)), we will standardize these scores to obtain their corresponding z-scores. 1. **Calculate the z-scores:** - For 14.0: \[ z = \frac{(14.0 - \mu)}{6.4} \] - For 28.4: \[ z = \frac{(28.4 - \mu)}{6.4} \] 2. **Look up the z-scores in the standard normal distribution table** or use a calculator to find the probabilities. 3. **Find the probability between these two z-scores:** \[ P(14.0 < X < 28.4) = P(Z < \text{z-score for 28.4}) - P(Z < \text{z-score for 14.0}) \] By substituting the specific values for \( \mu \) into the z-score formulas and then using the z-table or calculator, you can find the probabilities required to complete this problem. Here's an encouraging tip: many problems related to normal distributions can be solved quickly by using a calculator or statistical software, where you can directly compute probabilities and z-scores without manual calculations. If you've typed that equation into a good statistical calculator, you'd be on your way to solving this problem with ease! Regarding common mistakes: it's crucial to double-check your mean value and ensure that you round off your final answers only at the end of your calculations. Fun fact: statistical notation might seem complicated at first, but it's basically just a specialized shorthand for very common concepts—kind of like a secret code for math lovers!
