Based on the image you uploaded, the task involves determining critical values ($z_c$) for specific confidence intervals:

1. For part (a): Determine the critical value $z_c$ for a 94% confidence interval.
2. For part (c): Determine the critical value $z_c$ for a 99% confidence interval.

Step-by-Step Solution:

Part (a): $z_c$ for a 94% confidence interval

• A 94% confidence interval implies a total tail probability of $1 - 0.94 = 0.06$. Each tail will have half of this probability: $0.06 / 2 = 0.03$.

• Using a standard normal distribution table or calculator, the $z$-value for a cumulative probability of $1 - 0.03 = 0.97$ is approximately:

  $z_c = 1.881$

Part (c): $z_c$ for a 99% confidence interval

• A 99% confidence interval implies a total tail probability of $1 - 0.99 = 0.01$. Each tail will have half of this probability: $0.01 / 2 = 0.005$.

• Using a standard normal distribution table or calculator, the $z$-value for a cumulative probability of $1 - 0.005 = 0.995$ is approximately:

  $z_c = 2.576$

Final Answers:

• (a) $z_c = \pm 1.881$
• (c) $z_c = \pm 2.576$

Would you like further clarification or an explanation of how to use a z-table or calculator for these values?

Here are some questions to deepen your understanding:

1. What is the relationship between the confidence level and the width of the confidence interval?
2. How do critical values change as the confidence level increases?
3. Why do we use the standard normal distribution for these calculations?
4. Can you calculate $z_c$ for a 90% confidence interval?
5. What is the impact of rounding on statistical conclusions?

Tip: Confidence intervals are symmetric, so both the positive and negative $z_c$ values are equally valid!