Let's solve each of these scenarios using the normal and t-distributions.

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Normal Distribution

For a standard normal distribution, we use the $Z$-score:

1. Left-sided: $\text{InvNorm}(0.05) = -1.645$ This is the $Z$-score that leaves 5% of the area to the left.

2. Right-sided: $\text{InvNorm}(1 - 0.05) = \text{InvNorm}(0.95) = 1.645$ This is the $Z$-score that leaves 5% of the area to the right, or 95% of the area to the left.

3. Two-sided:

   • Left tail: $\text{InvNorm}\left(\frac{0.05}{2}\right) = \text{InvNorm}(0.025) = -1.96$
   • Right tail: $\text{InvNorm}\left(1 - \frac{0.05}{2}\right) = \text{InvNorm}(0.975) = 1.96$ These are the $Z$-scores that leave 2.5% in each tail (5% total).

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t-Distribution

For the $t$-distribution, with degrees of freedom $\text{df} = n - 1 = 49$, we use the $t$-score:

1. Left-sided: $\text{InvT}(0.05, 49) = -1.677$ This is the $t$-score that leaves 5% of the area to the left with 49 degrees of freedom.

2. Right-sided: $\text{InvT}(1 - 0.05, 49) = \text{InvT}(0.95, 49) = 1.677$ This is the $t$-score that leaves 5% of the area to the right, or 95% to the left, with 49 degrees of freedom.

3. Two-sided:

   • Left tail: $\text{InvT}\left(\frac{0.05}{2}, 49\right) = \text{InvT}(0.025, 49) = -2.009$
   • Right tail: $\text{InvT}\left(1 - \frac{0.05}{2}, 49\right) = \text{InvT}(0.975, 49) = 2.009$ These are the $t$-scores that leave 2.5% in each tail (5% total) with 49 degrees of freedom.

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Would you like additional explanations on these calculations or more details about interpreting the results?

Related Questions

1. How do you interpret the values obtained from InvNorm and InvT in hypothesis testing?
2. Why do we use the t-distribution instead of the normal distribution when the sample size is small?
3. How would these critical values change if we increased the sample size $n$?
4. What is the significance of using a two-sided test versus a one-sided test in statistical inference?
5. Can you explain the difference between a $Z$-score and a $t$-score in terms of their applications?

Tip

Remember, the t-distribution is generally wider (has heavier tails) than the normal distribution, especially for smaller sample sizes, because it accounts for additional uncertainty due to estimating the population standard deviation.