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?

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