In a recent​ year, the scores for the reading portion of a test were normally​ distributed, with a mean of 21.5 and a standard deviation of 5.3. Complete parts​ (a) through​ (d) below.
Question content area bottom
Part 1
​(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 17.
The probability of a student scoring less than 17 is 
  
​(Round to four decimal places as​ needed.)
Part 2
​(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 14.1 and 28.9.
The probability of a student scoring between 14.1 and 28.9 is 
  

​(Round to four decimal places as​ needed.)

We are working with a normal distribution here, where the mean (μ) is 21.5 and the standard deviation (σ) is 5.3. To solve these parts, we'll use the **Z-score formula** to convert the raw scores into Z-scores, and then we can find the corresponding probabilities using standard normal distribution tables or a calculator:

\[
Z = \frac{X - \mu}{\sigma}
\]

### Part (a) - Probability that a score is less than 17:

1. **Calculate the Z-score** for \( X = 17 \):

\[
Z = \frac{17 - 21.5}{5.3} = \frac{-4.5}{5.3} \approx -0.8491
\]

2. **Find the probability** corresponding to this Z-score. Using the standard normal distribution table or a calculator, the probability for \( Z = -0.8491 \) is approximately:

\[
P(Z < -0.8491) \approx 0.1985
\]

Thus, the probability that a randomly selected student scores less than 17 is approximately **0.1985**.

---

### Part (b) - Probability that a score is between 14.1 and 28.9:

1. **Calculate the Z-score for 14.1**:

\[
Z = \frac{14.1 - 21.5}{5.3} = \frac{-7.4}{5.3} \approx -1.3962
\]

2. **Calculate the Z-score for 28.9**:

\[
Z = \frac{28.9 - 21.5}{5.3} = \frac{7.4}{5.3} \approx 1.3962
\]

3. **Find the probabilities** for these Z-scores:
   - \( P(Z < -1.3962) \approx 0.0816 \)
   - \( P(Z < 1.3962) \approx 0.9184 \)

4. **Subtract the two probabilities** to find the probability that the score is between 14.1 and 28.9:

\[
P(14.1 < X < 28.9) = P(Z < 1.3962) - P(Z < -1.3962) = 0.9184 - 0.0816 = 0.8368
\]

Thus, the probability that a randomly selected student scores between 14.1 and 28.9 is approximately **0.8368**.

---

Would you like more details on this solution or further clarification?

### Related Questions:
1. What is the probability of scoring above 25 on the test?
2. How would you find the probability for a score less than 10?
3. What Z-score corresponds to the top 10% of the distribution?
4. If a student scores 30, what is their percentile rank?
5. What is the probability of a score between 20 and 22?

### Tip:
For normal distributions, knowing how to use Z-scores is crucial for finding probabilities and percentiles, especially when data doesn't match typical values from tables.

In a recent​ year, the scores for the reading portion of a test were normally​ distributed, with a mean of 21.5 and a standard deviation of 5.3. 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 that is less than 17.
(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 14.1 and 28.9.

Solve a problem involving a normal distribution with a mean of 21.5 and a standard deviation of 5.3. Learn how to calculate the probability of scores being less than 17 or between 14.1 and 28.9 using Z-scores. Suitable for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation