STAT 245 Lecture Notes - Lecture 6: Guesstimate, Sampling Distribution, Random Variable
STAT 245 – Exercises #6
Solutions
PROBLEM A: In filling bags with flour, the automatic filling process dispenses amounts that follow a
Normal distribution with a mean of 10.15 kg and a standard deviation of 0.12 kg. The label on the
bags indicates that the amount of flour in the bags is 10 kg.
[1]. If someone purchase one bag of this flour, what is the probability that the purchased bag contains
less than 10.10 kg of flour?
Let X be the random variable denoting the amount of flour in a bag. Then,
~ . , .X N 10 15 0 12
and
[2]. If over the course of one year a person purchases nine bags of this flour, what is the probability
that the average amount contained in the nine bags purchased is less than 10.10 kg of flour?
Let be the random variable denoting the average weight of flour in 9 bags. Then, by the result for the
sampling distribution of the mean of a random sample from a Normal population, we have
and
[3]. What is the probability that the total weight of the nine bags of flour purchased exceeds 90.90
kg?
The total weight of 9 bags of flour exceeds 90.90 kg if and only if the average weight of the 9 bags exceeds
. Hence,
PROBLEM B: The distribution of salaries of faculty at a particular university is highly skewed to the right,
averaging $87,654 with a standard deviation of $12,345.
[1]. Why is it impossible to determine the probability that the income of one randomly chosen faculty
member at this university exceeds $90,000?
The shape of the population distribution is not normal and the sample size is small, i.e. the sample consist
of only one respondent, and this means the Central Limit Theorem does not apply here and hence, we do
not know the shape of the sampling distribution. Therefore, we cannot determine the probability that
the income of one randomly chosen faculty member at this university exceeds $90,000.
[2]. What is the probability that the average income of a random sample of 50 faculty members at
this university exceeds $90,000?
Let be the random variable denoting the average income of a sample of 50 faculty members.
Since , that means the sample size is large i.e. and the Central Limit Theorem now applies.
The sampling distribution is approximately normal and we have
. Hence,
[3]. What result studied in class makes it possible to answer question [2] here? Give the proper name
of this result.
The Central Limit Theorem
PROBLEM C: How long does it take students who live off-campus and within walking distance of the
university to get to class each day? It has been claimed that the average time is 30 minutes with a
standard deviation of 10 minutes, but it has been suggested that a simple random sample of such
students should be obtained and a confidence interval for the mean time determined.
[1]. How large should the sample be in order to be able to estimate the true mean time such that the
estimate has a probability of 0.95 of being within 1.5 minutes of the true value?
We want to determine sample size n such that .
We want
