To solve this, we need to understand the relationships between the z-scores and their associated confidence levels in a normal distribution. Let’s break it down step by step.

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(a) Confidence level for $x \pm \frac{2.88\sigma}{\sqrt{n}}$

• The z-score is $z = 2.88$.
• Confidence level corresponds to the area under the normal curve between $-z$ and $z$. This is calculated as:

$\text{Confidence Level} = \text{Cumulative Area}(-2.88 \leq Z \leq 2.88)$

Using the standard normal table or a calculator:

$\text{Cumulative Probability at } z = 2.88 = 0.9980$

The confidence level is:

$\text{Confidence Level} = 0.9980 - (1 - 0.9980) = 99.6\%$

Answer: $99.6\%$

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(b) Confidence level for $x \pm \frac{1.41\sigma}{\sqrt{n}}$

• The z-score is $z = 1.41$.
• Confidence level corresponds to the cumulative area under the normal curve between $-1.41$ and $1.41$.

Using the standard normal table or a calculator:

$\text{Cumulative Probability at } z = 1.41 = 0.9207$

The confidence level is:

$\text{Confidence Level} = 0.9207 - (1 - 0.9207) = 92.1\%$

Answer: $92.1\%$

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(c) What value of $z_{\alpha/2}$ results in a 99.7% confidence level?

The confidence level is 99.7%, so the total area outside this range is:

$\alpha = 1 - 0.997 = 0.003$

Each tail will have:

$\alpha/2 = 0.0015$

We look up the z-score for the cumulative probability of $1 - 0.0015 = 0.9985$ in the standard normal table. This corresponds to:

$z_{\alpha/2} = 3.00$

Answer: $3.00$

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(d) What value of $z_{\alpha/2}$ results in a 62% confidence level?

The confidence level is 62%, so the total area outside this range is:

$\alpha = 1 - 0.62 = 0.38$

Each tail will have:

$\alpha/2 = 0.19$

We look up the z-score for the cumulative probability of $1 - 0.19 = 0.81$ in the standard normal table. This corresponds to:

$z_{\alpha/2} = 0.88$

Answer: $0.88$

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Final Answers:

(a) $99.6\%$
(b) $92.1\%$
(c) $3.00$
(d) $0.88$

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Would you like me to explain any step in more detail? Here are related questions you might consider:

1. How do we interpret z-scores in confidence intervals?
2. How do you calculate the cumulative area for non-standard z-scores?
3. How do confidence intervals change with increasing $z$-values?
4. What is the relationship between confidence levels and significance levels?
5. Why do we divide $\alpha$ by 2 in confidence interval calculations?

Tip: Always ensure your cumulative probabilities correspond to the desired confidence level to avoid calculation errors.