Question

The amount of syrup that people put on their pancakes is normally distributed with mean 59 mL and standard deviation 9 mL. Suppose that 18 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 X

? X

• ~ N(,)
• What is the distribution of ¯x
• ? ¯x
• ~ N(,)
• If a single randomly selected individual is observed, find the probability that this person consumes is between 60.5 mL and 62.1 mL.
• For the group of 18 pancake eaters, find the probability that the average amount of syrup is between 60.5 mL and 62.1 mL.
• For part d), is the assumption that the distribution is normal necessary? YesNo

Answer 1

a) The distribution of X here is given as:

b) Using Central limit theorem, the distribution of the sample mean here is computed as:

This is the required distribution here.

c) The probability here is computed as:

P( 60.5 < X < 62.1 )

Converting it to a standard normal variable, we get here:

Getting it from the standard normal tables, we get here:

Therefore 0.0685 is the required probability here.

d) Now the probability here is computed in same way but instead of standard deviation of 9, we use standard deviation in part b) as 2.1213

Getting it from the standard normal tables, we get here:

Therefore 0.1678 is the required probability here.

Yes, the assumption of normal distribution is necessary here because the sample size is not large enough.