Question

Question2 The data or information related to some populations and their samples are given as the followings (you can only use the partial given tables (Z, T, or Chi-Square) to find your answers. if you can’t find the exact values from the given tables, you should use the approximation approaches using the given tables only. We used similar approximations in our tutorials). If cases such as nonparametric methods or t' (approximation of t-student) are the only ways to solve certain questions, you should only mention the method. For Z, approximation of Z, or t-student cases, the complete solution is essential.

This assumption holds for Populations 1, 2, and 3: δ1 = δ2 = δ3

Population 1: N(μ1 ,δ12 )

Sample from Population 1: 2, 4, 8, 3, 9, 10, 11, 15, 19

Population 2: N(μ2 ,δ22 )

Sample from Population 2: 21, 14, 7, 13, 19, 20, 101, 35, 22

Population 3: N(μ3 ,δ32 )

Sample from Population 3: 11, 13, 7, 14, 10, 30, 13, 3, 20

Population 4: N(μ4 ,8 )

Sample from Population 4: 35, 10, 75, -11, 13, -50, 13, -3, 40, 17, 18, 20, 10, 100

Population 5: N(μ5 ,9 )

Sample from Population 5: 5, 10, 75, 10, 15, 37, 13, 47, 100, 15

Population 6: N(μ6 ,δ62 )

Sample from Population 6: 11, 13, 7, 14, 10, 30, 13, 3, 20, 28, 10, 34, 11, 13, 7, 14, 10, 30, 13, 3, 20, 28, 10, 34, 14, 10, 30, 13, 3, 20, 28, 10, 34, 11, 14, 10, 30, 13, 3, 20, 28, 10, 34, 11, 20, 28, 10, 34, 11, 14, 10, 30

Population 7: N(μ7 ,δ72 )

Sample from Population 7: 11, 13, 7, 14, 10, 30, 13, 3, 20, 40, 36, 28, 10, 34, 5, 10, 75, 10, 15, 37, 13, 47, 100, 15, 40, 36, 28, 10, 34, 5, 10, 75, 10, 15, 37, 13, 47, 100, 15, 40, 36, 28, 10, 34, 5, 10, 75, 10, 15, 37, 13, 47, 100, 15

Population 8: N(μ8 ,5 )

Sample from Population 8: 25, 10, 7, 14, 10, 30, 13, 3, 20, 60, 10, 34, 11, 13, 7, 14, 10, 30, 13, 3, 20, 28, 10, 34, 14, 10, 30, 13, 3, 20, 28, 10, 34, 11, 14, 10, 30, 13, 3, 20, 28, 10, 34, 11, 20, 28, 10, 34, 11, 14, 10, 30

Population 9: Unknown distribution

Sample from Population 9: 25, 10, 7, 14, 10, 30, 13, 3, 20, 60, 10, 34, 11, 13, 7, 14, 10, 30, 13, 3, 20, 28, 10, 34, 14, 10, 30, 13, 3, 20, 28, 10, 34, 11, 14, 10, 30, 13, 3, 20, 28, 10, 34, 11, 20, 28, 10, 34, 11, 14, 10, 30

Population 10: Unknown distribution

Sample from Population 10: 20, 28, 10, 34, 11, 14, 10, 30, 13, 3, 20, 28, 10, 34

Population 11: Unknown distribution

Sample from Population 11: 45, 30, 7, 14, 10, 30, 13, 3, 20, 60, 10, 34, 11, 13, 7, 14, 10, 30, 13, 3, 20, 28, 10, 11, 13, 7, 14, 10, 30, 13, 3, 20, 11, 14, 10, 30, 13, 3, 20, 28, 10, 34, 11, 20, 28, 10, 34, 11, 14, 10, 30, 70, 100

Population 12: Unknown distribution

Sample from Population 12: 13, 7, 14, 10, 30, 13, 3, 20, 11, 14, 10, 30, 13, 3, 20, 28, 10, 34

Can you please solve A to C in details. Thanks.

A Can we assume the difference between means of population 1 and 3 is equal to 1 with 99% confidence level? (Answer it using hypothesis testing with test-statistics)

B Can we assume the difference between means of population 6 and 7 is equal to 20 with 95% confidence level? (Answer it using hypothesis testing with p-value)

C Can we assume the difference between means of population 10 and 12 is more than 5 with 97% confidence level? (Answer it using hypothesis testing with test-statistic)

Answer 1

A Can we assume the difference between means of population 1 and 3 is equal to 1 with 99% confidence level? (Answer it using hypothesis testing with test-statistics)

The hypothesis being tested is:

H0: µ1 - µ2 = 1

H1: µ1 - µ2 ≠ 1

Group 1	Group 2
9.00	13.44	mean
5.61	7.80	std. dev.
9	9	n
	16	df
	-4.444	difference (Group 1 - Group 2)
	46.139	pooled variance
	6.793	pooled std. dev.
	3.202	standard error of difference
	1	hypothesized difference
	-1.700	t
	.1084	p-value (two-tailed)

The p-value is 0.1084.

Since the p-value (0.1084) is greater than the significance level (0.01), we fail to reject the null hypothesis.

Therefore, we can conclude that the difference between means of population 1 and 3 is equal to 1.

B Can we assume the difference between means of population 6 and 7 is equal to 20 with 95% confidence level? (Answer it using hypothesis testing with p-value)

The hypothesis being tested is:

H0: µ1 - µ2 = 20

H1: µ1 - µ2 ≠ 20

Group 1	Group 2
17.10	28.63	mean
9.64	24.88	std. dev.
52	54	n
	69	df
	-11.533	difference (Group 1 - Group 2)
	3.640	standard error of difference
	20	hypothesized difference
	-8.664	t
	1.22E-12	p-value (two-tailed)

The p-value is 0.0000.

Since the p-value (0.0000) is less than the significance level (0.05), we can reject the null hypothesis.

Therefore, we cannot conclude that the difference between means of population 6 and 7 is equal to 20.

C Can we assume the difference between means of population 10 and 12 is more than 5 with 97% confidence level? (Answer it using hypothesis testing with test-statistic)

The hypothesis being tested is:

Null hypothesis	H₀: η₁ - η₂ = 5
Alternative hypothesis	H₁: η₁ - η₂ > 5

Method	W-Value	P-Value
Not adjusted for ties	238.00	0.296
Adjusted for ties	238.00	0.295

The p-value is 0.296.

Since the p-value (0.296) is greater than the significance level (0.03), we fail to reject the null hypothesis.

Therefore, we cannot conclude that the difference between means of population 10 and 12 is more than 5.