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.H_{0}: At least two of the group means μ_{1}, μ_{2}, μ_{3}, μ_{4} are not equal.
H_{0}: μ_{1} = μ_{2} = μ_{3} = μ_{4}
Part (b)
State the alternative hypothesis.H_{a}: At least two of the group means μ_{1}, μ_{2}, μ_{3}, μ_{4} are not equal.
H_{a}: μ_{1} = μ_{2} = μ_{3} = μ_{4}
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.)If H_{0} 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 H_{0} 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 H_{0} 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 H_{0} 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.)reject the null hypothesisdo 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. Since α > p-value, we reject the null hypothesis.Since α < p-value, we do not reject the null hypothesis.
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.
H_{0}: μ_{1} = μ_{2} = μ_{3} = μ_{4}
H_{a}: At least two of the group means μ_{1}, μ_{2}, μ_{3}, μ_{4} 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 H_{0} 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.
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