Find the t* multiplier for a 99% confidence interval when the sample size is 45. Give your answer to 4 decimal places.

To find the \( t^* \) multiplier for a 99% confidence interval when the sample size is 45, we need to use the t-distribution. The \( t^* \) value corresponds to the critical value of the t-distribution for a two-tailed test with \(\alpha = 0.01\) (since 99% confidence means there's 1% left for the tails, and it's split into 0.5% for each tail). 

The degrees of freedom (df) for the t-distribution in this case are:

\[ \text{df} = n - 1 = 45 - 1 = 44 \]

Using a t-distribution table or statistical software to find the critical value for \(\alpha/2 = 0.005\) with 44 degrees of freedom:

The \( t^* \) value for \( \alpha/2 = 0.005 \) and df = 44 is approximately:

\[ t^* \approx 2.6844 \]

So, the \( t^* \) multiplier for a 99% confidence interval with a sample size of 45 is \( 2.6844 \).

Would you like further details or have any questions?

Here are 8 relative questions that expand on this topic:
1. How does the \( t^* \) value change as the sample size increases?
2. Why do we use the t-distribution instead of the normal distribution for small sample sizes?
3. What is the critical value \( t^* \) for a 95% confidence interval with 44 degrees of freedom?
4. How would the \( t^* \) value differ for a one-tailed test with the same confidence level?
5. Can you explain the relationship between degrees of freedom and the shape of the t-distribution?
6. How do we calculate the margin of error using the \( t^* \) value?
7. What assumptions must be met to use the t-distribution for constructing confidence intervals?
8. How does the width of a confidence interval change with different confidence levels?

**Tip:** Always ensure your sample data meets the assumptions required for using the t-distribution, such as normality of the data and independence of observations.

Learn how to find the t* multiplier for a 99% confidence interval when the sample size is 45. Understand the use of the t-distribution and critical values for statistical inference. Suitable for college-level students and professionals.

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