Question

"In a recent year, the scores for the reading portion of a test were normally distributed, with a mean of 22.8 and a standard deviation of 6.3. 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 less than 17.6. (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 between 18.5 and 27.1. (Round to four decimal places as needed.)

(c) Find the probability that a randomly selected high school student who took the reading portion of the test has a score more than 35.5. (Round to four decimal places as needed.)

Answer 1

To determine the probabilities of various score ranges on a reading test, we calculate Z-scores and use a standard normal distribution table or a statistics calculator. The Z-score is the number of standard deviations a score is from the mean, and the probabilities can be determined by looking up these Z-scores.

Step-by-step explanation:

To solve the problems stated, we will use the standard normal distribution and Z-scores, which indicate how many standard deviations an element is from the mean. A Z-score can be calculated using the formula Z = (X - μ) / σ, where X is the value in the distribution, μ is the mean, and σ is the standard deviation.

Problem (a)

Find the probability of a score less than 17.6.

Z = (17.6 - 22.8) / 6.3 = -0.8254

Using a Z-table or calculator, find P(Z < -0.8254).

Problem (b)

Find the probability of a score between 18.5 and 27.1.

Z1 = (18.5 - 22.8) / 6.3 = -0.6825
Z2 = (27.1 - 22.8) / 6.3 = 0.6825

Find P(Z1 < Z < Z2).

Problem (c)

Find the probability of a score more than 35.5.

Z = (35.5 - 22.8) / 6.3 = 2.0159

Find P(Z > 2.0159).

The required probabilities can be obtained by looking up the Z-scores on a normal distribution table or using a statistics calculator.