AMS 315 Data Analysis: Homework 1
Due: Jan 12, 2024 11:59pm
1. A biologist wishes to estimate the effect of an antibiotic on the growth of a particular bac-
terium by examining the mean amount of bacteria present per plate of culture when a fixed
amount of the antibiotic is applied. Previous experimentation with the antibiotic on this
type of bacteria indicates that the standard deviation of the amount of bacteria present is
approximately 15 cm2.
(a) Determine the number of observations (cultures that must be developed and then tested)
to estimate the mean amount of bacteria present, using a 99% confidence interval with
a half-width of 4 cm2.
(b) Suppose that the number of observation is 100, and the observed sample mean is 35
cm2. Construct a 95% confidence interval for µ. Assume that σ can be replaced by s.
2. A study was conducted with 100 adult male patients following a new treatment for congestive
heart failure. One of the variables measured on the patients was the increase in exercise
capacity (in minutes) over a 4-week treatment period. The previous treatment regime had
produced an average increase of µ = 2 minutes. The researchers wanted to evaluate whether
the new treatment had increased the value of µ in comparison to the previous treatment.
The data yielded y¯ = 2.15 and s = 1.1.
(a) Using α = 0.05, what conclusions can you draw about the research hypothesis?
(b) What is the probability of making a Type II error if the actual value of µ is 2.1?
3. A national agency sets recommended daily dietary allowances for many supplements. In par-
ticular, the allowance for zinc for males over the age of 50 years is 16 mg/day. The agency
would like to determine if the dietary intake of zinc for active males is significantly higher
than 16 mg/day. How many males would need to be included in the study if the agency
wants to construct an α = 0.05 test with the probability of committing a Type II error to
be at most 0.10 whenever the average zinc content is 16.3 mg/day or higher? Suppose from
previous studies they estimate the standard deviation to be approximately 5 mg/day.
4. A researcher uses a random sample of n = 17 items and obtain y¯ = 13.1, s = 2.8. Using an
α = 0.05 test, is there significant evidence in the data to support Ha : µ > 12? Place both
bounds on the level of significance of the test and rejection region based on the observed data.
5. Conduct a test of H0 : µ1 ≤ µ2 versus Ha : µ1 > µ2 for the sample data summarized here.
Use α = 0.05. Assume that we have normal populations with σ1 = σ2.
Group 1 2
Sample size 15 12
Sample mean 70.1 75.5
Sample Standard Deviation 7.23 4.68
1
Treatment 1 5.5 4.9 5.3 5.1 5.5 4.6 5.8
Treatment 2 4.3 3.8 4.7 3.9 5.3 4.4 5.2
6. Let ∆ = µ1 − µ2. Conduct a test of H0 : ∆ = 0 versus Ha : ∆ > 0 for the sample data given
here. Use α = 0.05.
7. Consider the paired data given here.
Pair 1 2 3 4 5 6
y1 48.2 41.3 47.4 50.6 43.5 49.1
y2 38.5 39.1 43.0 47.8 40.2 41.7
(a) Conduct a paired t test of H0 : µd ≤ 0 versus Ha : µd > 0 with d = y1 − y2. Use
α = 0.05.
(b) Use a testing procedure with binomial distribution, test the hypotheses in (a).
Hint: Try to compute p-value by binomial distribution under assumption that P (Y1 > Y2) =
0.5. And if P (Y1 > Y2) ≤ 0.5, µd ≤ 0. You will need to build a new T.S. to count the number
of observed pairs which are consistent with null hypothesis.
8. A study is being planned to evaluate the possible side effects of an anti-inflammatory drug.
It is suspected that the drug may lead to an elevation in the blood pressure of users of the
drug. A preliminary study of two groups of patients, one receiving the drug and the other
receiving a placebo, provides the following information on the systolic blood pressure (in mm
Hg) of the two groups:
Group Mean Standard Deviation
Placebo 125.0 19.5
Users of drug 129.5 19.7
Assume that both groups have systolic blood pressures that have a normal distribution with
standard deviations relatively close to the values obtained in the pilot study. Suppose the
study plan provides for the same number of patients in the placebo as in the treatment group.
Determine the sample size necessary for an α = 0.05 t test to have power of 0.90 to detect
an increase of 3 mm Hg in the blood pressure of the treatment group relative to the placebo
group.