11-4. Exercises

1. BASIC

Ex 1. The following three random samples are taken from three normal populations with respective means $μ_1$, $μ_2$ , and $μ_3$ , and the same variance $σ^2$ .

Sample 1	Sample 2	Sample 3
2	3	0
2	5	1
3	7	2
5		1
3

1. Find the combined sample size n.

2. Find the combined sample mean $\bar{x}$ .

3. Find the sample mean for each of the three samples.

4. Find the sample variance for each of the three samples.

5. Find $MST$ .

6. Find $MSE$.

7. Find $F=\frac{MST}{MSE}$ .

Ex 2. The following three random samples are taken from three normal populations with respective means $μ_1$, $μ_2$ , and $μ_3$ , and the same variance $σ^2$ .

Sample 1	Sample 2	Sample 3
0.0	1.3	0.2
0.1	1.5	0.2
0.2	1.7	0.3
0.1		0.5
		0.0

1. Find the combined sample size n.

2. Find the combined sample mean $\bar{x}$ .

3. Find the sample mean for each of the three samples.

4. Find the sample variance for each of the three samples.

5. Find $MST$ .

6. Find $MSE$.

7. Find $F=\frac{MST}{MSE}$ .

Ex 3. Refer to Exercise 1.

1. Find the number of populations under consideration K.

2. Find the degrees of freedom $df_1=K−1$ and $df_2=(n−K)$ .

3. For $α=0.05$ , find $F_α$ with the degrees of freedom computed above.

4. At $α=0.05$, test hypotheses $H_0:μ_1=μ_2=μ_3$ vs.  $H_a: at least one pair of the population means are not equal$

Ex 4. Refer to Exercise 2.

1. Find the number of populations under consideration K.

2. Find the degrees of freedom $df_1=K−1$ and $df_2=(n−K)$ .

3. For $α=0.01$ , find $F_α$ with the degrees of freedom computed above.

4. At $α=0.01$, test hypotheses $H_0:μ_1=μ_2=μ_3$ vs.  $H_a: at least one pair of the population means are not equal$

2. APPLICATIONS

Ex 5. The Mozart effect refers to a boost of average performance on tests for elementary school students if the students listen to Mozart’s chamber music for a period of time immediately before the test. In order to attempt to test whether the Mozart effect actually exists, an elementary school teacher conducted an experiment by dividing her third-grade class of 15 students into three groups of 5. The first group was given an end-of-grade test without music; the second group listened to Mozart’s chamber music for 10 minutes; and the third groups listened to Mozart’s chamber music for 20 minutes before the test. The scores of the 15 students are given below:

Group 1	Group 2	Group 3
80	79	73
63	73	82
74	74	79
71	77	82
70	81	84

Using the ANOVA F-test at $α=0.10$ , is there sufficient evidence in the data to suggest that the Mozart effect exists?

Ex 6. The Mozart effect refers to a boost of average performance on tests for elementary school students if the students listen to Mozart’s chamber music for a period of time immediately before the test. Many educators believe that such an effect is not necessarily due to Mozart’s music per se but rather a relaxation period before the test. To support this belief, an elementary school teacher conducted an experiment by dividing her third-grade class of 15 students into three groups of 5. Students in the first group were asked to give themselves a self-administered facial massage; students in the second group listened to Mozart’s chamber music for 15 minutes; students in the third group listened to Schubert’s chamber music for 15 minutes before the test. The scores of the 15 students are given below:

Group 1	Group 2	Group 3
79	82	80
81	84	81
80	86	71
89	91	90
86	82	86

Test, using the ANOVA F-test at the 10% level of significance, whether the data provide sufficient evidence to conclude that any of the three relaxation method does better than the others.

Ex 7. Precision weighing devices are sensitive to environmental conditions. Temperature and humidity in a laboratory room where such a device is installed are tightly controlled to ensure high precision in weighing. A newly designed weighing device is claimed to be more robust against small variations of temperature and humidity. To verify such a claim, a laboratory tests the new device under four settings of temperature-humidity conditions. First, two levels of high and low temperature and two levels of high and low humidity are identified. Let T stand for temperature and H for humidity. The four experimental settings are defined and noted as (T, H): (high, high), (high, low), (low, high), and (low, low). A pre-calibrated standard weight of 1 kg was weighed by the new device four times in each setting. The results in terms of error (in micrograms mcg) are given below:

(high, high)	(high, low)	(low, high)	(low, low)
−1.50	11.47	−14.29	5.54
−6.73	9.28	−18.11	10.34
11.69	5.58	−11.16	15.23
−5.72	10.80	−10.41	−5.69

Test, using the ANOVA F-test at the 1% level of significance, whether the data provide sufficient evidence to conclude that the mean weight readings by the newly designed device vary among the four settings.

Ex 8. To investigate the real cost of owning different makes and models of new automobiles, a consumer protection agency followed 16 owners of new vehicles of four popular makes and models, call them $TC$ , $HA$ , $NA$ , and $FT$ , and kept a record of each of the owner’s real cost in dollars for the first five years. The five-year costs of the 16 car owners are given below:

TC	HA	NA	FT
8423	7776	8907	10333
7889	7211	9077	9217
8665	6870	8732	10540
	7129	9747
	7359	8677

Test, using the ANOVA F-test at the 5% level of significance, whether the data provide sufficient evidence to conclude that there are differences among the mean real costs of ownership for these four models.

Ex 9. Helping people to lose weight has become a huge industry in the United States, with annual revenue in the hundreds of billion dollars. Recently each of the three market-leading weight reducing programs claimed to be the most effective. A consumer research company recruited 33 people who wished to lose weight and sent them to the three leading programs. After six months their weight losses were recorded. The results are summarized below:

Statistic	Prog. 1	Prog. 2	Prog. 3
Sample Mean	$\bar{x_1}=10.65$	$\bar{x_2}=8.90$	$\bar{x_3}=9.33$
Sample Variance	$s_1^2=27.20$	$s_2^2=16.86$	$s_3^2=32.40$
Sample Size	$n_1=11$	$n_2=11$	$n_3=11$

The mean weight loss of the combined sample of all 33 people was $\bar{x}=9.63$. Test, using the ANOVA F-test at the 5% level of significance, whether the data provide sufficient evidence to conclude that some program is more effective than the others.

Ex 10. A leading pharmaceutical company in the disposable contact lenses market has always taken for granted that the sales of certain peripheral products such as contact lens solutions would automatically go with the established brands. The long-standing culture in the company has been that lens solutions would not make a significant difference in user experience. Recent market research surveys, however, suggest otherwise. To gain a better understanding of the effects of contact lens solutions on user experience, the company conducted a comparative study in which 63 contact lens users were randomly divided into three groups, each of which received one of three top selling lens solutions on the market, including one of the company’s own. After using the assigned solution for two weeks, each participant was asked to rate the solution on the scale of 1 to 5 for satisfaction, with 5 being the highest level of satisfaction. The results of the study are summarized below:

Statistics	Sol. 1	Sol. 2	Sol. 3
Sample Mean	$\bar{x_1}=3.28$	$\bar{x_2}=3.96$	$\bar{x_3}=4.10$
Sample Variance	$s_1^2=0.15$	$s_2^2=0.32$	$s_3^2=0.36$
Sample Size	$n_1=18$	$n_2=23$	$n_3=22$

The mean satisfaction level of the combined sample of all 63 participants was $\bar{x}=3.81$. Test, using the ANOVA F-test at the 5% level of significance, whether the data provide sufficient evidence to conclude that not all three average satisfaction levels are the same.

3. LARGE DATA SET EXERCISE

10KB

data9.xls

Ex 11. Large Data Set 9 records the costs of materials (textbook, solution manual, laboratory fees, and so on) in each of ten different courses in each of three different subjects, chemistry, computer science, and mathematics. Test, at the 1% level of significance, whether the data provide sufficient evidence to conclude that the mean costs in the three disciplines are not all the same.