Section 7.1 – Distribution of Sample Proportions for samples of size n. (p-hat distribution)
1) Singular is a medication to control asthma attacks. There is a claim that 18.4% of the patients who
take Singular experience headaches as a side effect. However, other company claims that actually
more than 18.4% of patients experience headache.
To test the 18.4% claim, a sample of 400 patients taking Singular was selected at random, and it was
observed that 93 of them experienced headaches as a side effect.
Let’s study the distribution of sample proportions (for p = 0.184) for samples of size 400 and observe
how this sample fits in such a distribution
a) Describe the sampling distribution: Assuming that the 18.4% claim is correct, give the shape, mean and
standard deviation of the distribution of sample proportions for samples of size 400. Sketch and label
b) Find the likelihood of this sample statistic: What is the probability that at least 93 patients experience
headaches in a random sample of 400 patients who use this medication?
 Find the sample proportion, place it on the graph above and shade the required area (How does this
statistic “fit” on this distribution? Is it usual or unusual; likely to observe or unlikely?)

Show all steps to find the probability

Check with the calculator feature
c) Interpret results from part (b).
If 18.4% of patients who take Singular experience headache, there is a _______% probability of observing 93 or
more patients experiencing headaches in groups of 400.
This means that if we take 1000 samples of size 400, about _______ of those samples will show 93 or more
people experiencing headaches.
This is a _________________ event if 18.4% of patients experience headache as a side effect. (Complete with one
of the following choices)
Very likely
likely
unlikely
very unlikely
d) Infer: What may this result suggest about the percentage of people who take Singular and experience
headaches?
Is it lower than 18.4%, equal to 18.4% or higher than 18.4%?
Section 7.1 – Distribution of Sample Proportions for samples of size n. SOLUTIONS
2) Singular is a medication to control asthma attacks. There is a claim that 18.4% of the patients who
take Singular experience headaches as a side effect. However, other company claims that actually
more than 18.4% of patients experience headache.
To test the 18.4% claim, a sample of 400 patients taking Singular was selected at random, and it was
observed that 93 of them experienced headaches as a side effect.
Let’s study the distribution of sample proportions (for p = 0.184) for samples of size 400 and observe
how this sample fits in such a distribution
a) Describe the sampling distribution: Assuming that the 18.4% claim is correct, give the shape, mean and
standard deviation of the distribution of sample proportions for samples of size 400. Sketch and label
n = 400, p = .184 (Since np and n(1-p) are both greater than 15, the p-hat distribution can be
approximated with a normal distribution)
p-hat ~ App.N(   p  .184 ,  
p
p
You sketch the graph and label
p (1 p )
.184*.816

 .0193742)
n
400
b) Find the likelihood of this sample statistic: What is the probability that at least 93 patients experience
headaches in a random sample of 400 patients who use this medication?
 Find the sample proportion, place it on the graph above and shade the required area (How does this
statistic “fit” on this distribution? Is it usual or unusual; likely to observe or unlikely?)
x 93
p 
 .2325 . Since p-hat is more than two standard errors to the right
n = 400, x = 93,
n 400
of the mean it’s an unusual sample statistic.
 Show all steps to find the probability
Calculate the z-score corresponding to a p-hat = .2325
p p
.2325.184
z

 2.5
.0193742
p (1 p )
n
Now calculate the probability:
P(p-hat > .2325) = P(z > 2.5) = 1 - .9938 = .0062 ~ 0.006 = 6/1000 = 3/500
 Check with the calculator feature
Normalcdf(.2325, 10^9, .184, .019374) = .00615
c) Interpret results from part (b).
 If 18.4% of patients who take Singular experience headache, there is a _6% probability of observing 93
or more patients experiencing headaches in groups of 400.
 This means that if we take 1000 samples of size 400, about __6_ of those samples will show 93 or more
people experiencing headaches.
 This is a _very unlikely_ event if 18.4% of patients experience headache as a side effect. (Complete with
one of the following choices)
Very likely
likely
unlikely
very
unlikely
d) Infer: What may this result suggest about the percentage of people who take Singular and experience
headaches?
Is it lower than 18.4%, equal to 18.4% or higher than 18.4%?This may suggest that probably the
percentage of people who take Singular and experience headaches as a side effect is higher than 18.4%