Question

1.When students apply for graduate studies (i.e. at the master’s or doctoral level), they are
required to submit an official copy of their transcript, mailed directly from the Registrar’s
Office at their academic institution. The customer service division of the Registrar’s
Office at a large Canadian university is interested in determining if they are more than
25% faster at processing transcripts than another university in the area, which can process
transcripts in 16 business hours. The customer service manager obtains a random sample
of 10 waiting times (in business hours), which are provided below.
11 12 18 20 23
15 10 12 13 14
a. Conduct an appropriate hypothesis test. Use the critical value method. Use a
population standard deviation of 2 hours. [9 marks]
HINT: You will first have to determine what it means to be 25% faster, in terms
of hours.
b. Explain what a Type I Error means in this context. [1 mark]   
2. A major keyboard manufacturer has a line of keyboards designed for apartment dwellers.
These keyboards need to be light enough to be carried up flights of stairs. The lead
engineer wants to use a new type of material. The engineer claims that the new keyboards
will be lighter than the old keyboards.
They take a sample of 4 keyboards manufactured using the old material and compute an
average weight of 21 kg with a standard deviation of 1 kg.
They take a sample of 8 keyboards manufactured using the new material and compute an
average weight of 17 kg with a standard deviation of 2 kg.
a. Conduct an appropriate hypothesis test using the p-value method. Use the old
material as population 1. [8 marks]
b. How much evidence is there against the null hypothesis in part (a)? [1 mark]
c. Explain what a Type II Error means in this context. [1 mark

3.An insurance company is interested in estimating the population mean cost of basic dental
cleaning at dentists in Saskatoon. Suppose there are only two dentists in Saskatoon:
Dentist A and Dentist B. Suppose also that the cost of basic dental cleaning varies only
depending on how well the patient practices regular dental hygiene, so that the cost of
basic dental cleaning roughly follows a Normal distribution regardless of the dentist.
The insurance company selects 8 sample patients and sends them to both Dentist A and
Dentist B. They send the patients in random order, such that half of the patients are seen
by Dentist A first, and half are seen by Dentist B first, so as not to bias the results. The
cost of basic dental cleaning for these 8 patients seen by both Dentists A and B are
provided below. The insurance company would like to determine whether the population
mean cost of basic dental cleaning by Dentist A is different from the population mean
cost of basic dental care by Dentist B. Let the population of costs of basic dental care
from Dentist A be population 1.
Patient 1 2 3 4 5 6 7 8
Dentist A $100 $120 $125 $110 $95 $105 $120 $115
Dentist B $150 $100 $140 $100 $95 $105 $100 $120
Conduct an appropriate hypothesis test using the critical value method. [10 marks]

Answer 1

1.

a)

25% =0.25

New time =Old time/(1+0.25) =16/1.25 =12.8 hours.

So, more than 25% faster than 16 hours means processing transcripts in less than 12.8 hours.

Since the sample size, n =10 <30 (small sample), we shall use t-score.

One sample t-test:

Null Hypothesis, H0: 12.8 hours.

Alternative Hypothesis, H1: < 12.8 hours. (left-tailed test).

( =Population mean time; =Hypothesised population mean time =12.8 hours).

Sample mean, =(11+12+.......+14)/10 =14.8

Given: population standard deviation, =2 hours.

Test statistic, t =(​​​​​​)/(​​​​​​) =(14.8 - 12.8)/(2/​​​​​​) =3.1623

Degrees of freedom, df =n-1 =10-1 =9

Let the significance level (alpha) =5% =0.05

For a left-tailed test, at 0.05 alpha, at df =9, the critical value of t is: t_crit = -1.8331

Decision criteria: Since it is a left-tailed test, reject H0 if t < t_crit.

Conclusion: Since t: 3.1623 > t_crit: -1.8331, we failed to reject the null hypothesis(H0) at 5% significance level. Thus, we do not have a sufficient statistical evidence to claim that they are more than 25% faster at processing transcripts than another university in the area which can process transcripts in 16 business hours.

b)

Type I Error is rejecting the true null hypothesis. In this context, Type I Error means - even if it is true that they are not more than 25% faster, we incorrectly reject it and claim that they are more than 25% faster at a set significance level which is the probability of type I error.