STAT 381 - Summer Assignment Solutions

Name________________________________
STAT 381 – Summer
Must show work for partial credit. Even if you used technology to calculate something,
give a formula or some indication of what you did to calculate it.
1. The breaking strength of a random sample of 20 bundles of wool fibers have a sample
mean of 436.5 and a sample standard deviation of 11.90. A normal probability plot
provides strong support for assuming that the population distribution of breaking
strength is at least approximately normal. Construct a 95% confidence interval for the
average breaking strength. Put your conclusion in the context of the problem.
(10
points)
Answer
Given n=20 , x =436.5 , σ =11.90;
Construct a 95% confidence interval for the average bvreaking strength.
Formula for 95% confidence interval is given by;
P ( X−
Z α
2
∗σ
√ n < μ< X +
Z α
2
∗σ
√ n )
¿ X ± Zα /2
σ
√ n
Here we take alpha = 0.05 as the level of significance so the Z(alpha) = Z(0.05) =
1.96
So we have the confidence as follows;
¿ ( X ± Zα /2∗σ
√n )
¿ (436.5 ± 1.96∗11.90
√20 )
¿ (436.5−1.96∗11.90
√20 , 436.5+ 1.96∗11.90
√20 )
¿ ( 436.5−5.2154 , 436.5+5.2154 )
¿ ( 431.2846 , 441.7154 )
From the above calculations, we are 95% confident that the true population mean lies
between 431.2846 and 441.7154.
Conclusion: We can conclude that if a sample lies in the interval (431.2846,
441.7154) then we accept the null hypothesis.
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2. A milk container is supposed to contain 2 Liters of milk, but 10% of the time, it is
underweight. The milk containers are shipped to retail outlets in boxes of 20
containers. (10 points)
A. Let
X be the number of underweight containers in a box. Explain why
X is a
binomial random variable.
Answer
Let x be the no. of underweight container.
The probability that a container is fixed for each container at 10% (p = 0.1).
No. of tails are fixed at 20 →n = 20.
Therefore, X has binomial distribution with parameters n=20 and p=0.1
B. What is the probability that exactly three of the twenty containers are
underweight? (On this part, write out the formula, even if you evaluate with
technology).
Answer
P( Exactly three)=P(x=3)
P ( X=x ) = Cx
n ∗px∗(1−p)n− x
P ( X=3 )= Cx
n ∗px∗( 1− p )n−x= C3
20 ∗0.13∗( 1−0.3 )20−3=1140∗0.001∗0.1668
P ( X=3 )=0.1902
C. What is the probability that at most three of the twenty containers are
underweight?
Answer
Probability that at most three of the 20 containers are underweight P(X = 3):
P ( Atmost three )=P ( x ≤ 3 ) =P ( X=0 )+ P ( X =1 ) + P ( x =2 )+ P ( X=3 )
P ( x ≤ 3 ) = [ C0
20 ∗0.10∗( 1−0.3 ) 20−0
] + [ C1
20 ∗0.11∗( 1−0.3 ) 20−1
] + [ C2
20 ∗0.12∗( 1−0.3 ) 20−2
] + [ C3
20 ∗0.13∗( 1−0.3 ) 20−3
]
P ( x ≤ 3 ) =0.1216+ 0.2702+ 0.2852+ 0.1901=0.8671
P ( x ≤ 3 )=0.8671
D. What is the expected value and variance of the number of containers that will be
underweight?
Answer
Expected value E(x) is given as follows;
E ( X ) =np=20∗0.1=2
Variance value V ar ( x ) is given as follows;
Var ( X )=np ( 1− p )=20∗0.1∗(1−0.1)=1.8
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3. A new drug is being compared to a standard drug for treating a particular illness.
In the clinical trials, a group of 200 patients was randomly split into two groups,
with one group being given the standard drug and one given the new drug.
Altogether, 83 out of the 100 patients given the new drug improved their
condition, while only 72 out of the 100 patients given the standard drug improved
their condition. Conduct a hypothesis test with a level of significance of .05 to
investigate whether the new drug is better than the standard drug. Comment on
appropriateness of the method (based on size) and put conclusion in the context of
the problem. (15 points)
A. Explain why the comparison of two population proportions is the correct method:
Answer
In order to compare two population proportions, the two samples to be tested
should be randomly be drawn and be independent of each other. Each of the
sample should have at least 5 cases (sample size). In this case, we had the two
samples having 83 and 72 cases respectively and as such we can conclude that
method is correct.
B. Use the recommended sequence of steps:
1. Identify the parameter of interest and describe it in the context of the problem
situation:
Answer
Parameter of interest are as follows;
 Population proportion of patients whose conditions improved , after giving
new drug (denoted by p1)
 Population proportion of patients whose conditions improved , after giving
standard drug (denoted by p2)
2. Determine the null value and state the null hypothesis:
Answer
The null value is zero for the difference between the two samples ( p1− p2=0)
The null hypothesis can be written as follows;
H0 : p1− p2=0
3. State the appropriate alternative hypothesis:
Answer
Alternative hypothesis is given as follows;
H A : p1− p2 >0
4. Give the formula for the computed value of the test statistic (substituting the
null value and the known values of any other parameters, but not those of any
sample-based quantities):
Answer
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