CHAPTER 7.2 – DISTRIBUTION OF SAMPLE MEANS (x-bar distribution) – typical questions
1) Describe the distribution of sample means for samples of size n
a) Give shape, explain why it is normal or not
b) Give the mean
c) Give the standard deviation (standard error)
d) Between what two values will most of the sample means lie
2) Find probabilities in the x-bar distribution
3) Does the observed sample mean make you question the population mean stated in the problem? Why?
4) What effect does increasing the sample size have in the x-bar distribution?
PROBLEM:
The reading speed of second grade students is approximately normal, with a mean  = 91 words per minute
(wpm) and a standard deviation of  = 10 wpm. In a random sample of 25 students the observed mean
reading speed was 94 words per minute.
a) Describe the distribution of sample means for samples of size 25. Shape, mean and standard error.
b) Between what two values will most of the sample means lie? Sketch, label.
c) What is the probability that a random sample of 25 second grade students results in a mean reading rate
of more than 94 words per minute? Use the sketch to part (b) to shade. Show all work to answer.
d) Do these results make you doubt the mean of 91 stated in the problem? (Is the reading speed of 94 an
unusual sample mean? Yes, No, Explain why: (i) based on z-scores, (ii) based on probabilities.)
e) What is the calculator input to answer question (c).
f) Use your calculator to simulate the experiment of selecting 25 numbers from the given normal
distribution, and finding the mean of the selected numbers. Repeat 10 times and use your results to find
the experimental probability that x-bar is more than 94. randNorm(mean, stdev, 25) →L1:meanL1
g) How does the experimental probability compare with the theoretical probability of part (c)? Comment
on the law of large numbers.
h) I just realize that the problem has a typo. In reality, the mean reading speed of 94 words per minute was
observed in a random sample of 55 students (instead of 25 as stated above). How does this affect the
results of the problem?
CHAPTER 7.1 – DISTRIBUTION OF SAMPLE PROPORTIONS (p-hat distribution) – typical questions
5) Describe the distribution of sample proportions for samples of size n
a) Give shape, explain why it is normal or not
b) Give the mean
c) Give the standard deviation (standard error)
d) Between what two values will most of the sample means lie
6) Find probabilities in the x-bar distribution
7) Does the observed sample proportion make you question the population mean stated in the problem?
Why?
8) What effect does increasing the sample size have in the p-hat distribution?
PROBLEM:
According to the Fall 2012 MC Student Profile, 35.0% of students in the college are 18-20 years old.
A sample of 100 students has been selected at random with 40% of them being 18-20 years old.
a) Describe the distribution of sample proportions for samples of size 100. Shape, mean and standard
error.
b) Between what two values will most of the sample proportions lie? Sketch, label.
c) What is the probability that in a random sample of 100 MC students more than 40% of them are 1820 years old? Use the sketch to part (g) to shade. Show all work to answer.
d) Do these results make you doubt the population proportion of 0.35 stated in the problem? (Is the
sample proportion of 0.40 an unusual sample proportion? Yes, No, Explain why: (i) based on zscores, (ii) based on probabilities.
e) What is the calculator input to answer question (c).
f) Use your calculator to simulate the experiment of selecting 100 numbers from a binomial
distribution with a probability of success of 0.35. randBin (n, p). Find the sample proportion p-hat.
Repeat 10 times and use your results to find the experimental probability that p-hat is more than 0.40
g) How does the experimental probability compare with the theoretical probability of part (c)?
Comment on the law of large numbers.
h) Assume that in a random sample of 144 students, 40% of them are 18-20 years old. How does this
affect the results of the problem?