Question

The authors of a paper studied a random sample of 346 Twitter users. For each Twitter user in the sample, the tweets sent during a particular time period were analyzed and the Twitter user was classified into one of the following categories based on the type of messages they usually sent.

Category	Description
IS	Information sharing
OC	Opinions and complaints
RT	Random thoughts
ME	Me now (what I am doing now)
O	Other

The accompanying table gives the observed counts for the five categories (approximate values read from a graph in the paper).

Twitter Type	IS	OC	RT	ME	O
Observed count	51	60	62	99	74

Carry out a hypothesis test to determine if there is convincing evidence that the proportions of Twitter users falling into each of the five categories are not all the same. Use a significance level of α = 0.05.

Let p₁, p₂, p₃, p₄, and p₅ be the proportions of Twitter users falling into the five categories.

State the appropriate null and alternative hypotheses.

H₀: p₁ = p₂ = p₃ = p₄ = p₅ = 0.5
H_a: H₀ is not true.

H₀: p₁ = p₂ = p₃ = p₄ = p₅ = 346
H_a: H₀ is not true.

H₀: p₁ = p₂ = p₃ = p₄ = p₅ = 0.2
H_a: H₀ is not true.

H₀: p₁ = p₂ = p₃ = p₄ = p₅ = 70
H_a: H₀ is not true.

H₀: p₁ = p₂ = p₃ = p₄ = p₅ = 0.05
H_a: H₀ is not true.

Find the test statistic and P-value. (Use technology. Round your test statistic to three decimal places and your P-value to four decimal places.)

X²=

P-value=

State the conclusion in the problem context.

Do not reject H₀. There is convincing evidence to conclude that the proportions of Twitter users falling into the five categories are not all the same.

Do not reject H₀. There is not convincing evidence to conclude that the proportions of Twitter users falling into the five categories are not all the same.

Reject H₀. There is not convincing evidence to conclude that the proportions of Twitter users falling into the five categories are not all the same.

Reject H₀. There is convincing evidence to conclude that the proportions of Twitter users falling into the five categories are not all the same.

Answer 1

Claim: There is convincing evidence that the proportions of Twitter users falling into each of the five categories are not all the same.

p = 1 / 5 = 0.2

The null and alternative hypothesis is

H₀: p₁ = p₂ = p₃ = p₄ = p₅ = 0.2
H_a: H₀ is not true.

Test statistic is

O: Observed frequency
E: Expected frequency.

E = n*p

n = 346

	O	E	(O-E)	(O-E)^2	(O-E)^2/E
	51	69.2	-18.2	331.24	4.786705
	60	69.2	-9.2	84.64	1.223121
	62	69.2	-7.2	51.84	0.749133
	99	69.2	29.8	888.04	12.83295
	74	69.2	4.8	23.04	0.332948
Total	346				19.925

Degrees of freedom = Number of E's - 1 = 5 - 1 = 4

P-value = P() = 0.0005

P-value < 0.05 we reject null hypothesis.

Reject H₀. There is convincing evidence to conclude that the proportions of Twitter users falling into the five categories are not all the same.