The amount of coffee that people drink per day is normally distributed with a mean of 16 ounces and a standard deviation of 6.5 ounces. 34 randomly selected people are surveyed. Round all answers to 4 decimal places where possible. a. What is the distribution of $X ? X \sim N(16$ 6.5 $\sigma^{8}$ $\sigma^{4}$ b. What is the distribution of $\bar{x} ? \bar{x} \sim \mathrm{N}(16$ $\sigma^{s}$ c. What is the probability that one randomly selected person drinks between 15.6 and 16.1 ounces of coffee per day? d. For the 34 people, find the probability that the average coffee consumption is between 15.6 and 16.1 ounces of coffee per day. e. For part d), is the assumption that the distribution is normal necessary? $\bigcirc$ NoO Yes f. Find the IQR for the average of 34 coffee drinkers. \[ \begin{array}{ll} Q 1=\square & \text { ounces } \\ Q 3=\square & \text { ounces } \\ I Q R: & \\ & \text { ounces } \end{array} \]

Step 1 ：The distribution of X is $X \sim N(16, 6.5^2)$

Step 2 ：The distribution of the sample mean $\bar{x}$ is $\bar{x} \sim N(16, (6.5^2)/34)$

Step 3 ：To find the probability that one randomly selected person drinks between 15.6 and 16.1 ounces of coffee per day, we need to standardize these values and use the standard normal distribution table. The standardized values are $Z_1 = (15.6 - 16)/6.5 = -0.0615$ and $Z_2 = (16.1 - 16)/6.5 = 0.0154$. The probability is $P(Z_1 < Z < Z_2) = P(-0.0615 < Z < 0.0154)$

Step 4 ：To find the probability that the average coffee consumption of 34 people is between 15.6 and 16.1 ounces of coffee per day, we need to standardize these values using the standard deviation of the sample mean and use the standard normal distribution table. The standardized values are $Z_1 = (15.6 - 16)/\sqrt{(6.5^2)/34} = -0.3348$ and $Z_2 = (16.1 - 16)/\sqrt{(6.5^2)/34} = 0.0842$. The probability is $P(Z_1 < Z < Z_2) = P(-0.3348 < Z < 0.0842)$

Step 5 ：For part d), the assumption that the distribution is normal is necessary because we are dealing with a sample mean and we are using the Central Limit Theorem

Step 6 ：To find the IQR for the average of 34 coffee drinkers, we need to find the first quartile $Q1$ and the third quartile $Q3$ of the distribution of the sample mean. $Q1$ and $Q3$ are the 25th and 75th percentiles of the distribution, respectively. We can find these values using the standard normal distribution table and then convert them back to the scale of the sample mean. The IQR is $Q3 - Q1$