Questions: The test scores for the analytical writing section of a particular standardized test can be approximated by a normal distribution, as shown in the figure. (a) What is the maximum score that can be in the bottom 5% of scores? (b) Between what two values does the middle 90% of scores lie? (a) The maximum score that can be in the bottom 5% is (Round to two decimal places as needed.)

The test scores for the analytical writing section of a particular standardized test can be approximated by a normal distribution, as shown in the figure.
(a) What is the maximum score that can be in the bottom 5% of scores?
(b) Between what two values does the middle 90% of scores lie?
(a) The maximum score that can be in the bottom 5% is (Round to two decimal places as needed.)

Transcript text: The test scores for the analytical writing section of a particular standardized test can be approximated by a normal distribution, as shown in the figure. (a) What is the maximum score that can be in the bottom $5 \%$ of scores? (b) Between what two values does the middle $90 \%$ of scores lie? (a) The maximum score that can be in the bottom $5 \%$ is $\square$ (Round to two decimal places as needed.)

Solution

Solution Steps

Step 1: Find the z-score corresponding to the bottom 5%.

We want to find the z-score such that the area to the left of it is 0.05. Looking up a z-table or using a calculator, we find that the z-score corresponding to 0.05 is approximately -1.645.

Step 2: Calculate the score.

We use the z-score formula: z = (x - μ) / σ. We are given μ = 3.7 and σ = 0.94. We want to find x, the score. Plugging in the values, we have -1.645 = (x - 3.7) / 0.94.

Step 3: Solve for x.

Multiply both sides by 0.94: -1.645 * 0.94 = x - 3.7. -1.5463 = x - 3.7 x = 3.7 - 1.5463 x ≈ 2.15