1. **Problem 47:** Given that $P(T_{18} > t) = 0.975$, find $t$. 2. The t-distribution with 18 degrees of freedom is symmetric and centered at zero. 3. The probability $P(T_{18} > t) = 0.975$ means the area to the right of $t$ under the t-distribution curve is 0.975. 4. Since the total area under the curve is 1, the area to the left of $t$ is $1 - 0.975 = 0.025$. 5. From t-distribution tables or using statistical software, the critical value $t$ for $P(T_{18} ext{ less than } t) = 0.025$ is approximately $-2.1009$. 6. Therefore, $t = -2.1009$, which corresponds to option (b). --- 7. **Problem 50:** Comparing the t-distribution with the standard normal distribution: 8. The t-distribution has heavier tails than the normal distribution, meaning it is more likely to produce values far from the mean. 9. It is also less peaked in the center compared to the normal distribution. 10. Hence, the correct description is: "The t-distribution is less peaked in the center and has higher tails," which corresponds to option (b).