Question

The following table lists the number of pages in four different types of magazines.

Home decorating	News	Health	Computer
175	89	84	109
287	97	154	139
165	124	91	101
208	107	107	208
200	104	98	149

Using a significance level of 5%, test the hypothesis that the four magazine types have the same average length. (Let 1 = Home decorating, 2 = News, 3 = Health and 4 = Computer.)

• Part (a)

  State the null hypothesis.

  H₀: At least two of the group means μ₁, μ₂, μ₃, μ₄ are not equal.

  H₀: μ₁ = μ₂ = μ₃ = μ₄

      

• Part (b)

  State the alternative hypothesis.

  H_a: At least two of the group means μ₁, μ₂, μ₃, μ₄ are not equal.

  H_a: μ₁ = μ₂ = μ₃ = μ₄

      

• Part (c)

  Enter an exact number as an integer, fraction, or decimal.
  df(num) =

• Part (d)

  Enter an exact number as an integer, fraction, or decimal.
  df(denom) =

• Part (e)

  State the distribution to use for the test.

  F_{3, 16}

  F_{3, 19}

      

  F_{19, 16}

  F_{16, 3}

  F_{16, 19}

• Part (f)

  What is the test statistic? (Round your answer to two decimal places.)

• Part (g)

  What is the p-value? (Round your answer to four decimal places.)
  
  
  Explain what the p-value means for this problem.

  If H₀ is true, then there is a chance equal to the p-value that the value of the test statistic will be equal to or less than the calculated value.If H₀ is true, then there is a chance equal to the p-value that the value of the test statistic will be equal to or greater than the calculated value.    If H₀ is false, then there is a chance equal to the p-value that the value of the test statistic will be equal to or greater than the calculated value.If H₀ is false, then there is a chance equal to the p-value that the value of the test statistic will be equal to or less than the calculated value.

• Part (h)

  Sketch a picture of this situation. Label and scale the horizontal axis, and shade the region(s) corresponding to the p-value.

• Part (i)

  Indicate the correct decision ("reject" or "do not reject" the null hypothesis) and write appropriate conclusions.(i) Alpha (Enter an exact number as an integer, fraction, or decimal.)
  α =
  
  (ii) Decision:

  reject the null hypothesisdo not reject the null hypothesis    

  
  (iii) Reason for decision:

  Since α < p-value, we reject the null hypothesis.Since α > p-value, we do not reject the null hypothesis.    Since α > p-value, we reject the null hypothesis.Since α < p-value, we do not reject the null hypothesis.

  
  (iv) Conclusion:

  There is sufficient evidence to conclude that the four magazine types do not have the same average length.There is not sufficient evidence to conclude that the four magazine types do not have the same average length.

Answer 1

H₀: μ₁ = μ₂ = μ₃ = μ₄

H_a: At least two of the group means μ₁, μ₂, μ₃, μ₄ are not equal.

c(

df(num) = k(groups) -1=3

d)

df(denom) = n-k=20-4 =16

e)

F_{3, 16}

f)

Applying one way ANOVA (from excel: data-data analysis)

Source of Variation	SS	df	MS	F	P-value
Between Groups	34370.80	3	11456.93	9.0854	0.0010
Within Groups	20176.40	16	1261.03
Total	54547.20	19

test statistic =9.09

p value =0.0010

If H₀ is true, then there is a chance equal to the p-value that the value of the test statistic will be equal to or greater than the calculated value.  

i)\

reject" the null hypothesis)
α = 0.05

reject the null hypothesis

  Since α > p-value, we reject the null hypothesis

There is sufficient evidence to conclude that the four magazine types do not have the same average length.