Question

assume a standard normal distribution. draw a well-labeled normal curve for each part.

a. find the z-score that gives a left area of 0.7002.
b. find the z-score that gives a left area of 0.1668.

a. z = 0.52 (round to two decimal places as needed.)

choose the correct answer below.

Explanation:

⚡ Using what you learned: normal distribution

Step 1: Identify the z-score for part (a)

We need to find the \( z \)-score corresponding to a cumulative left area of \( 0.7002 \).

Using a standard normal distribution table (or calculator):

$$ P(Z < z) = 0.7002 \implies z \approx 0.52 $$

Step 2: Select the correct graph for part (a)

• The \( z \)-score is \( 0.52 \), which is a positive value located slightly to the right of the center (\( z = 0 \)).
• The "left area" represents the region shaded to the left of this \( z \)-score.
• Since the area is \( 0.7002 \) (which is greater than \( 0.5 \)), more than half of the curve must be shaded to the left of the boundary line.

Comparing the options:

• A: Shaded to the left of a negative \( z \)-score.
• B: Shaded to the right of a positive \( z \)-score.
• C: Shaded to the left of a positive \( z \)-score (\( z \approx 0.52 \)), covering more than half of the curve.
• D: Shaded to the right of a negative \( z \)-score.

Thus, graph C is the correct representation.

Step 3: Find the z-score for part (b)

We need to find the \( z \)-score corresponding to a cumulative left area of \( 0.1668 \).

Using a standard normal distribution table:

$$ P(Z < z) = 0.1668 \implies z \approx -0.97 $$

Answer:

a. \( z = 0.52 \)
Graph Choice: C
b. \( z = -0.97 \)