H ONLY # Study Duration in the Library AnalysisTo analyze the amount of time students spend studying in the library, we consider a scenario where students' study times are normally distributed with a mean of 42 minutes and a standard deviation of 23 minutes. A sample size of 10 students is observed. Let's go through the detailed statistical questions and their solutions:### a. Distribution of Individual Study Times**Question:** What is the distribution of $X$?**Answer:** $X \sim N(42, 23)$This indicates that the individual study times follow a normal distribution with a mean (μ) of 42 minutes and a standard deviation (σ) of 23 minutes.### b. Distribution of Sample Mean Study Time**Question:** What is the distribution of $\bar{x}$ for a sample of 10 students?**Answer:** $\bar{x} \sim N(42, 7.2733)$Here, $\bar{x}$ represents the sample mean study time, which follows a normal distribution with a mean of 42 minutes and a standard deviation of $\frac{23}{\sqrt{10}} \approx 7.2733$ minutes.### c. Distribution of Total Study Time**Question:** What is the distribution of $\sum x$ for a sample of 10 students?**Answer:** $\sum x \sim N(420, 72.732)$The total study time ($\sum x$) for 10 students follows a normal distribution with a mean of 10 * 42 = 420 minutes and a standard deviation of $23 \times \sqrt{10} \approx 72.732$ minutes.### d. Probability of Individual Study Time Between 35 and 45 Minutes**Question:** What is the probability that one randomly selected student's study time is between 35 and 45 minutes?**Answer:** 0.1714Using the normal distribution, the probability calculation for this interval yields a result of approximately 0.1714.### e. Probability of Average Study Time for 10 Students Between 35 and 45 Minutes**Question:** What is the probability that the average time studying for 10 students is between 35 and 45 minutes?**Answer:** 0.4921For the sample mean distribution, the probability that the average study time falls within this range is approximately 0.4921.

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Suppose that the amount of time that students spend studying in the library in one sitting is normally distributed with mean 42 minutes and standard deviation 23 minutes. A researcher observed 10 students who entered the library to study. Round all answers to 4 decimal places where possible. a. What is the distribution of X? X - N 42 23 0° 0 b. What is the distribution of ? N 42 7.2733 0° c. What is the distribution of ? Σ - N( 420 ✓ 72.732) or or d. If one randomly selected student is timed, find the probability that this student's time will be between 35 and 45 minutes. 0.1714 e. For the 10 students, find the probability that their average time studying is between 35 and 45 minutes. 0.4921 0 f. Find the probability that the randomly selected 10 students will have a total study time less than 360 minutes. 0.2047 08 g. For part e) and f), is the assumption of normal necessary? Yes No o h. The top 15% of the total study time for groups of 10 students will be given a sticker that says "Great dedication". What is the least total time that a group can study and still receive a sticker? 495.3508 X minutes OT