Questions: Find the z-scores that separate the middle 77% of the distribution from the area in the tails of the standard normal distribution. The z-scores are (Use a comma to separate answers as needed. Round to two decimal places as needed.)

Solution Steps

We need to find the \( z \)-scores that separate the middle \( 77\% \) of the standard normal distribution from the tails. This means that \( 23\% \) of the distribution is in the tails, with \( 11.5\% \) in each tail.

The probabilities for the tails are:

• Lower tail: \( P(Z < z_{\text{lower}}) = 0.115 \)
• Upper tail: \( P(Z < z_{\text{upper}}) = 1 - 0.115 = 0.885 \)

Using the standard normal distribution, we find the \( z \)-scores corresponding to these probabilities:

• For the lower tail: \( z_{\text{lower}} \) corresponds to \( P(Z < z_{\text{lower}}) = 0.115 \)
• For the upper tail: \( z_{\text{upper}} \) corresponds to \( P(Z < z_{\text{upper}}) = 0.885 \)

The calculated \( z \)-scores are:

• \( z_{\text{lower}} \approx -1.2 \)
• \( z_{\text{upper}} \approx 1.2 \)

Both \( z \)-scores are rounded to two decimal places:

• \( z_{\text{lower}} = -1.20 \)
• \( z_{\text{upper}} = 1.20 \)

Final Answer