To construct a 90% confidence interval, you'll typically need the following information:

1. The sample mean ($\bar{x}$)
2. The standard deviation ($s$) or population standard deviation ($\sigma$), if available
3. The sample size ($n$)
4. The confidence level (which is 90% in this case)

Formula for a confidence interval for the mean:

If you're dealing with a sample and the population standard deviation is unknown, use the t-distribution:

$\text{CI} = \bar{x} \pm t_{\alpha/2} \cdot \frac{s}{\sqrt{n}}$

Where:

• $\bar{x}$ is the sample mean
• $s$ is the sample standard deviation
• $n$ is the sample size
• $t_{\alpha/2}$ is the critical value from the t-distribution for a 90% confidence level and $n - 1$ degrees of freedom

If the population standard deviation ($\sigma$) is known, use the z-distribution:

$\text{CI} = \bar{x} \pm z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}}$

Where $z_{\alpha/2}$ is the critical value from the standard normal distribution for a 90% confidence level.

Steps to calculate:

1. Find the sample mean ($\bar{x}$).
2. Determine the standard deviation ($s$ or $\sigma$).
3. Find the appropriate critical value ($t_{\alpha/2}$ or $z_{\alpha/2}$).
   • For a 90% confidence level, the critical value $z_{\alpha/2}$ from the z-distribution is approximately 1.645.
   • If using the t-distribution, look up the t-value based on your sample size.
4. Calculate the margin of error: $t_{\alpha/2} \cdot \frac{s}{\sqrt{n}}$ or $z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}}$.
5. Construct the confidence interval by adding and subtracting the margin of error from the sample mean.

Please provide the sample mean, standard deviation, and sample size (if known), so I can assist you in calculating the confidence interval.

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Do you have these details, or would you like further clarification on any step?

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Related Questions:

1. What happens to the width of the confidence interval as the confidence level increases?
2. How does the sample size affect the margin of error in a confidence interval?
3. Why is the t-distribution used when the population standard deviation is unknown?
4. How do we determine the appropriate critical value for a confidence interval?
5. What role does variability in data play in constructing confidence intervals?

Tip: Increasing your sample size reduces the margin of error, leading to a narrower confidence interval.