An agronomist examines the cellulose content

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.
¯

(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 circumstance to assume that the sample mean sales X will be approximately normal?

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 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?