Question

The amount of syrup that people put on their pancakes is normally distributed with mean 55 mL and standard deviation 11 mL. Suppose that 17 randomly selected people are observed pouring syrup on their pancakes. Round all answers to 4 decimal places where possible.

1. What is the distribution of XX? XX ~ N(,)
2. What is the distribution of ¯xx¯? ¯xx¯ ~ N(,)
3. If a single randomly selected individual is observed, find the probability that this person consumes is between 53.5 mL and 56.5 mL.
4. For the group of 17 pancake eaters, find the probability that the average amount of syrup is between 53.5 mL and 56.5 mL.
5. For part d), is the assumption that the distribution is normal necessary? YesNo

Answer 1

Let X be the amount of syrup put on the pancakes by a randomly selected person. We know that X is normally distributed with mean and standard deviation

a) The distribution of X can be written as

ans:

b) Let be the sample mean of a randomly selected sample of size n=17. Since we know the population standard deviation, using the central limit theorem, we can say that is normally distributed with mean and standard deviation

The distribution of is

ans:

c) If a single randomly selected individual is observed, the probability that this person consumes is between 53.5 mL and 56.5 mL is same as the probability that X is between 53.5 mL and 56.5 mL

ans: If a single randomly selected individual is observed, the probability that this person consumes is between 53.5 mL and 56.5 mL is 0.1114

d) For the group of 17 pancake eaters, the probability that the average amount of syrup is between 53.5 mL and 56.5 mL. is same as the probability that is between 53.5 mL and 56.5 mL

ans: For the group of 17 pancake eaters, the probability that the average amount of syrup is between 53.5 mL and 56.5 mL. is 0.4246

e) Although the sample size n is less than 30, we know the population standard deviation.  Using the central limit theorem, we can say that is normally distributed, irrespective of the distribution of X.

the assumption that the distribution of X or   is normal not necessary

ans: No