Question

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]

NOTE: You are encouraged to use Excel to calculate the sample mean(s) and sample

standard deviation(s) for this question. If you use Excel for this, provide the entire

command (e.g. if you take the average of 1,2,3, then write down =AVERAGE(1,2,3)).

Round average and standard deviation calculations to 2 decimal places.

Answer 1

Solution:

Here, we have to use paired t test for the difference between population means.

Null hypothesis: H₀: The population mean cost of basic dental cleaning by Dentist A is not different from the population mean cost of basic dental care by Dentist B.

Alternative hypothesis: H_a: 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.

H₀: µ_d = 0 versus H_a: µ_d ≠ 0

This is a two tailed test.

We assume level of significance = α = 0.05

Test statistic for paired t test is given as below:

t = (Dbar - µ_d)/[S_d/sqrt(n)]

From given data, we have

Dbar = -2.50

S_d = 22.6779

n = 8

df = n – 1 = 7

α = 0.05

Critical Values = -2.3646 and 2.3646

(by using t-table)

t = (Dbar - µ_d)/[S_d/sqrt(n)]

t = (-2.50 - 0)/[ 22.6779/sqrt(8)]

t = -0.3118

Here, we have

-2.3646 < t < 2.3646

So, we do not reject the null hypothesis

There is insufficient evidence to conclude that 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.