AP Statistics
Z tests and intervals
1. The makers of Aspro brand aspirin want to be sure that their tablets contain the right amount
of active ingredient (acetylsalicylic acid). They inspect a random sample of 36 tablets from a
lot of production. When the production process is working properly, Aspro tablets have an
average of m = 320mg acetylsalicylic acid with a standard deviation of s = 3mg . Here are
the amounts (in mg) of acetylsalicylic acid in the 36 randomly selected tablets.
319
321
319
328
320
321
321
322
322
324
318
319
322
326
326
320
316
320
324
316
324
321
326
320
320
325
318
324
322
317
320
321
316
318
322
319
Q. Estimate the mean acetylsalicylic acid content of the tablets in this lot. (To answer this
question, construct a confidence interval (use a 90% confidence level). Use notebook paper,
there is not enough room on this paper.)
Step 1: The variable of interest is the mg of acetylsalicylic acid in Aspro tablets, using a zinterval for means.
Step 2: We have a random sample, given in the problem, and based on the sample size n = 36 >
30, by the CLT, the sampling distribution is approximately normal.
Step 3:
Step 4: We are 90% confident that the true mean acetylsalicylic acid content is between 320.2
and 321.9 mg. In repeated random sampling, this method successfully captures the true
mean 90% of the time.
Q. Should the company distribute these tablets to drugstores or dispose of the entire batch?
Justify your answer. Since the intended mean is within the interval, we have evidence that
the mean is OK, but this is not proof that the aspirin is OK. It might be reasonable to repeat
the test before taking action.
2. In the past, the mean score of the seniors in Prince Edward County on the American College
Testing (ACT) college entrance examination has been 20. This year a special preparation
course is offered, and all seniors planning to take the ACT test enroll in the course. The mean
of a random sample of 53 ACT scores is 22.1. Assume that ACT scores vary normally with
standard deviation 6. The principal believes that the new course has improved the students’
ACT scores. Is this so?
H :m = 20
Step 1: 0
Ha:m > 20
Step 2: We have a random sample, given in the problem, and based on the sample size n = 53 >
30, by the CLT, the sampling distribution is approximately normal.
Step 3: z =
x - m0
s
n
=
22.1 - 20
= 2.55
6
53
Step 4:
Step 5: P( Z > 2.548 ) = .0054
Step 6: A test statistic as large as this may occur by chance alone less than 1% of the time, so we
reject H0.
Step 7: We have strong evidence that the true mean SAT Math score in Prince Edward County is
greater than 20, so we can support the principal’s claim.
3. Previously we had a test of a hypothesis about SAT Math scores of California high school
students based on an SRS of 500 students. The hypotheses are H0 : m = 450; H a : m > 450 .
Assume that the population standard deviation is s = 100 . The test rejects H 0 at the 1% level of
significance. Is this test sufficiently sensitive to usually detect an increase of 10 points in the
population mean SAT Math score? Answer this question by calculating the power of the test
against the alternative m = 460 . Critical value:
(
invNorm .99, 450,100 / 500
)
= 460.40 = x
\Reject H 0 when x ³ 460.40 .
P ( x > 460.4 m = 460 ) = .4643 = power
This test is not sufficiently sensitive to detect an
increase of 10 points on the mean SAT Math score.
The power is less than .5.
4. A company medical director failed to find significant evidence that the mean systolic blood
pressure of population of executives differed from the national mean of 128. The medical
director now wonders if the test used would detect an important difference if one were
present. For the SRS of size 72 from a population with a standard deviation s = 15 , a twosided test of H0 : m = 128 is planned for a 5% significance level.
a. Find the power of the test against the alternative 134.
P( x > 131.5 ) = .92
b. Find the power of the test against the alternative 122. Can the test be relied on to detect a
mean that differs from 128 by 6?
This test is quite powerful
against alternatives 6 units away
from 128.
c. If the alternative were farther
away from H 0 , say m = 136 ,
would the power be higher or
lower than the values found above? higher