#5. Test of hypothesis:
Book problems, section 9.1: 1, 3, 6, 8, 11, 13, 14
1. Consider the hypothesis H0: God exists. State the appropriate alternative hypothesis, (b) What is
a type II error, (b) what is a type II error?
2. Which of the following statements about hypothesis testing are true?
a. A type I error occurs if H0 is rejected when it is true
b. A type II error occurs if H0 is rejected when it is true
c. Type I errors are always worse than type II errors.
A only,
b only,
c only,
a and c only
b and c only
3. You are given the following about a random sample: Y=X1+X2+…+Xn where the sample size n=25,
and the random variables are i.i.d. Each Xi has the Poisson distribution with parameter λ. H0:
λ=0.1, H1: λ<0.1. The critical region is C={Y≤3}. Calculate the significance level of the test.
4. You are given a random sample from a normal loss distribution: the sample mean is 42,000, the
sample standard deviation is 8,000, there are 25 loss observations in the sample. Using a two
sided test with H0: µ=45,000 and H1: µ≠45,000, at what level of α would you reject the null
hypothesis. A) α=.01, b) does not reject at α=.01, but reject at α=.02, c) does not reject at α=.02,
but reject at α=.05, d) does not reject at α=.05, but reject at α=.1, e) do not reject at α=.1.
5. A random sample of 21 observations from a normal distribution yields the following result:
sample mean=3.5, sample variance=0.6156, n=21. You are testing the hypothesis H0: µ=3 vs H1:
µ≠3. Calculate the p-value.
6. A survey claims that the average cost of a hotel room in Tulsa Oklahoma is less than $49.21. In
order to test this claim, a researcher selects a sample of 26 hotel rooms and finds an average
cost equal to 47.37, with a sample standard deviation of 6.42. Assume the population random
variable to be normal. At α=.05 comment on the claim.
7. Drivers are classified as “good” or “bad”. Results of the classification are assumed to be
binomially distributed with probability of being “good” equal to p. A sample consists of 100
drivers. Determine the critical value for testing the hypothesis p<0.5 with significance level α of
at most 0.05 using normal approximation.
8. Aaron was shooting his Koosh basketball into the hoop above his door. He claims that he is an
80% shooter. In a sample Aaron made 25 of 36 shoots. Comment on his claim at the 5% level.
9. X1,X2,…,X11 is a random sample of size 11 from a normal distribution with mean µ1 and variance
σ12, with a sample mean=2, and sample variance=4. Y1,Y2,…,Y13 is a random sample of size 13
from a normal distribution with mean µ2 and variance σ22, with a sample mean=1.5, and sample
variance=5. H0: µ1= µ2, vs H1: µ1> µ2. The variance is unknown but it is assumed to be the same
between the samples. Calculate the test statistic to evaluate the null hypothesis and compute
the p-value.
10. X1,X2,…,X11 is a random sample of size 11 from a normal distribution with mean µ1 and variance
σ12, with a sample mean=10, and sample variance=4. Y1,Y2,…,Y11 is a random sample of size 11
from a normal distribution with mean µ2 and variance σ22, with a sample mean=9, and sample
variance=12. H0: µ1= µ2, vs H1: µ2> µ1. The underlying variance is assumed to be the same for
both distributions. Let t be the sample t-statistic used to test the difference of the means from
the two normally distributed random samples. Let T be the critical value at α=.05 . Calculate
|t-T|.
11. X1,X2,…,X8 is a random sample where X is normally distributed, the sample mean =4, and sample
variance=16. Y1,Y2,…,Y9 is a random sample, where Y is normally distributed, with a sample
mean=3, and sample variance=9. H0: σ12= σ22, vs H1: σ12> σ22. Calculate the test statistic f, and the
critical value F, then find |f-F|.
12. The independent random variables X,Y are from separate normal distributions. Random samples
from each distribution are shown below:
X
Y
1
5
9
10
17
36
1
4
5
5
15
Using the sample data and a 5% significance level, which of the following are true statements
regarding the hypothesis testing?
a) Reject H0: σX2=50 in favor of Ha: σX2>50
b) Reject H0: σy2=50 in favor of Ha: σY2<50
c) Reject H0: σX2= σy2 in favor of Ha: σX2> σy2
I. a) only
II. b) only
III. a) and b)
IV. a) and c)
V. b) and c)