The following data are from a completely randomized design. In the following calculations, use ? = 0.05. Treatment 1 Treatment 2 Treatment 3 62 82 70 46 71 53 53 87 60 39 64 45 xj 50 76 57 sj2 96.67 108.67 112.67 (a) Use analysis of variance to test for a significant difference among the means of the three treatments. State the null and alternative hypotheses. H0: ?1 ≠ ?2 ≠ ?3 Ha: ?1 = ?2 = ?3H0: ?1 = ?2 = ?3 Ha: Not all the population means are equal. H0: Not all the population means are equal. Ha: ?1 = ?2 = ?3H0: ?1 = ?2 = ?3 Ha: ?1 ≠ ?2 ≠ ?3H0: At least two of the population means are equal. Ha: At least two of the population means are different. Find the value of the test statistic. (Round your answer to two decimal places.) Find the p-value. (Round your answer to three decimal places.) p-value = State your conclusion. Do not reject H0. There is sufficient evidence to conclude that the means of the three treatments are not equal.Reject H0. There is sufficient evidence to conclude that the means of the three treatments are not equal. Do not reject H0. There is not sufficient evidence to conclude that the means of the three treatments are not equal.Reject H0. There is not sufficient evidence to conclude that the means of the three treatments are not equal. (b) Use Fisher's LSD procedure to determine which means are different. Find the value of LSD. (Round your answer to two decimal places.) LSD = Find the pairwise absolute difference between sample means for each pair of treatments. x1 − x2 = x1 − x3 = x2 − x3 = Which treatment means differ significantly? (Select all that apply.) There is a significant difference between the means for treatments 1 and 2.There is a significant difference between the means for treatments 1 and 3.There is a significant difference between the means for treatments 2 and 3.There are no significant differences.

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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The following data are from a completely randomized design. In the following calculations, use

? = 0.05.

	Treatment 1	Treatment 2	Treatment 3
	62	82	70
46	71	53
53	87	60
39	64	45
x_j	50	76	57
s_j²	96.67	108.67	112.67

(a)

Use analysis of variance to test for a significant difference among the means of the three treatments.

State the null and alternative hypotheses.

H₀: ?₁ ≠ ?₂ ≠ ?₃
H_a: ?₁ = ?₂ = ?₃H₀: ?₁ = ?₂ = ?₃
H_a: Not all the population means are equal. H₀: Not all the population means are equal.
H_a: ?₁ = ?₂ = ?₃H₀: ?₁ = ?₂ = ?₃
H_a: ?₁ ≠ ?₂ ≠ ?₃H₀: At least two of the population means are equal.
H_a: At least two of the population means are different.

Find the value of the test statistic. (Round your answer to two decimal places.)

Find the p-value. (Round your answer to three decimal places.)

p-value =

State your conclusion.

Do not reject H₀. There is sufficient evidence to conclude that the means of the three treatments are not equal.Reject H₀. There is sufficient evidence to conclude that the means of the three treatments are not equal. Do not reject H₀. There is not sufficient evidence to conclude that the means of the three treatments are not equal.Reject H₀. There is not sufficient evidence to conclude that the means of the three treatments are not equal.

(b)

Use Fisher's LSD procedure to determine which means are different.

Find the value of LSD. (Round your answer to two decimal places.)

LSD =

Find the pairwise absolute difference between sample means for each pair of treatments.

x₁ − x₂

x₁ − x₃

x₂ − x₃

Which treatment means differ significantly? (Select all that apply.)

There is a significant difference between the means for treatments 1 and 2.There is a significant difference between the means for treatments 1 and 3.There is a significant difference between the means for treatments 2 and 3.There are no significant differences.

Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...

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