An agronomist examines the cellulose content

MATH 1131 3.00 A S1
Assignment 4
Total marks = 60
Question 1: Here are the weights (in kilograms) of a random sample of 24
male runners
67.8 61.9 63.0 53.1 62.3 59.7 55.4 58.9
60.9 69.2 63.7 68.3 64.7 65.6 56.0 57.8
66.0 62.9 53.6 65.0 55.8 60.4 69.3 61.7
Suppose that the standard deviation of the population of weights is known
to be σ = 4.5 kg.
(a) (4 marks). Stating any additional assumptions, give a 95% confidence
interval for the population mean weight µ. Are you quite sure that µ
is less than 65 kg? Why?
(b) (3 marks). How many male runners would you have needed to estimate
µ with a margin of error equal to 1 kg with 95% confidence?
(c) (2 marks). Based on the confidence interval you gave in part (a), does
a test of
H0 : µ = 61.3 kg
Ha : µ = 61.3 kg
reject H0 at the 5% significance level? Why?
(d) (2 marks). Would H0 : µ = 65 be rejected at the 5% level if tested
against a two-sided alternative? Why?
Question 2: An agronomist examines the cellulose content of a variety of
alfalfa hay. Suppose that the cellulose content in the population has standard
deviation σ = 8 milligrams per gram (mg/g). A sample of 15 cuttings has
mean cellulose content x = 145 mg/g.
¯

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(a) (2 marks). Give a 90% confidence interval for the population mean
cellulose content µ.
(b) (7 marks). A previous study claimed that the mean cellulose content
was µ = 140 mg/g, but the agronomist believes that the mean is higher
than that figure. State H0 and Ha and carry out a significance test to
see if the new data support this belief.
(c) (2 marks). The statistical procedures used in (a) and (b) are valid
when certain assumptions are satisfied. What are these assumptions?
Question 3: You want to see if a redesign of the cover of a mail-order
catalogue will increase the mean sales µ. A random sample of 900 customers
are to receive the catalogue with the new cover. For planning purposes,
you are willing to assume that the sales from the new catalogue will be
approximately normally distributed with σ = 50 dollars. The mean sales for
the original catalogue is 25 dollars. You wish to test
H0 : µ = 25
Ha : µ > 25
You decide to reject H0 if the sample mean sales x > 26 and to accept H0
¯
otherwise.
(a) (2 marks). Find the probability of a Type I error, that is, the probability that your test rejects H0 when in fact µ = 25 dollars.
(b) (2 marks). Find the probability of a Type II error when µ = 28 dollars.
This is the probability that your test accepts H0 when in fact µ = 28.
(c) (2 marks). Find the probability of a Type II error when µ = 30.
(d) (2 marks). The distribution of sales is not normal because many customers buy nothing. Why is it nonetheless reasonable in this circum¯
stance to assume that the sample mean sales X will be approximately
normal?

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Question 4: A company medical director failed to find significant evidence
that the mean blood pressure of a population of executives differed from
the national mean µ = 128. The medical director now wonders if the test
used would detect an important difference if one were present. For a random
sample of size 72 from a normal population of executive blood pressures with
standard deviation σ = 15, the z statistic is
√
z = (¯ − 128)/(15/ 72)
x
The two-sided test rejects
H0 : µ = 128
at the 5% level of significance when |z| ≥ 1.96.
(a) (5 marks). Find the power of the test against the alternative µ = 135.
(b) (4 marks). Find the power of the test against µ = 121. Can the test
be relied on to detect a mean that differs from 128 by 7?
(c) (1 mark). If the alternative were farther from H0 , say µ = 138, would
the power be higher or lower than the values calculated in (a) and (b)?
Question 5: (10 marks). Assume rents of one-bedroom apartments are
normally distributed. A random sample of 10 one-bedroom apartments from
your local newspaper has these monthly rents (in dollars):

500

650

600

505

450

550

515

495

650

395

Do these data give reason to believe that the mean rent of all advertised
apartments is greater than $500 per month? State hypotheses, find the t
statistic and its p-value, and state your conclusion.
Question 6: (2 marks). Large trees growing near power lines can cause
power failures during storms when their branches fall on the lines. Power
companies spend a great deal of time and money trimming and removing
trees to prevent this problem. Researchers are developing hormone and
chemical treatments that will stunt or slow tree growth. If the treatment
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is too severe, however, the tree will die. In one series of laboratory experiments on 216 sycamore trees, 41 trees died. Give a 95% confidence interval
for the proportion of sycamore trees that would be expected to die from this
particular treatment.
Question 7: (2 marks). A national opinion poll found that 44% of all American adults agree that parents should be given vouchers good for education
at any public or private school of their choice. The result was based on a
small sample. How large a random sample is required to obtain a margin of
error of ±0.03 (i.e., ±3%) in a 95% confidence interval? (use the previous
poll’s result to obtain the guessed value p∗ .)
Question 8: (6 marks). Of the 500 respondents in a Christmas tree telephone survey, 44% had no children at home and 56% had at least one child at
home. The corresponding figures for the most recent census are 48% with no
children and 52% with at least one child. Test the null hypothesis that the
telephone survey technique has a probability of selecting a household with
no children that is equal to the value obtained by the census. Give the z
statistic and the p-value. What do you conclude?