Question

The authors of a paper studied a random sample of 355 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 | 53 | 59 | 66 | 103 | 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 0.05.

Let *p*_{1}, *p*_{2},
*p*_{3}, *p*_{4}, and
*p*_{5} be the proportions of the five message types
among the population of Twitter users.

State the null and alternative hypotheses.

*H*_{0}: *p*_{1} =
*p*_{2} = *p*_{3} =
*p*_{4} = *p*_{5} = 0.5

*H*_{a}: *H*_{0} is not
true.

*H*_{0}: *p*_{1} =
*p*_{2} = *p*_{3} =
*p*_{4} = *p*_{5} = 355

*H*_{a}: *H*_{0} is not
true.

*H*_{0}: *p*_{1} =
*p*_{2} = *p*_{3} =
*p*_{4} = *p*_{5} = 70

*H*_{a}: *H*_{0} is not
true.

*H*_{0}: *p*_{1} =
*p*_{2} = *p*_{3} =
*p*_{4} = *p*_{5} = 0.2

*H*_{a}: *H*_{0} is not
true.

*H*_{0}: *p*_{1} =
*p*_{2} = *p*_{3} =
*p*_{4} = *p*_{5} = 0.05

*H*_{a}: *H*_{0} is not
true.

Calculate the test statistic. (Round your answer to two decimal
places.)

*χ*^{2} =

What is the *P*-value for the test? (Round your answer to
four decimal places.)

*P*-value =

What can you conclude?

Do not reject *H*_{0}. There is not enough
evidence to conclude that the proportions of Twitter users falling
into the five categories are not all the same.

Do not reject *H*_{0}. There is convincing
evidence to conclude that the proportions of Twitter users falling
into the five categories are not all the
same.

Reject *H*_{0}. There is convincing evidence to
conclude that the proportions of Twitter users falling into the
five categories are not all the same.

Reject *H*_{0}. There is not enough evidence to
conclude that the proportions of Twitter users falling into the
five categories are not all the same.

Answer #1

The authors of a paper studied a random sample of 353 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...

The authors of a paper concerned about racial stereotypes in
television counted the number of times that characters of different
ethnicities appeared in commercials aired on a certain city's
television stations, resulting in the data in the accompanying
table.
Ethnicity
African-
American
Asian
Caucasian
Hispanic
Observed Frequency
60
10
324
6
Based on the 2000 Census, the proportion of the U.S. population
falling into each of these four ethnic groups are 0.177 for
African-American, 0.032 for Asian, 0.734 for Caucasian,...

The authors of a paper concerned about racial stereotypes in
television counted the number of times that characters of different
ethnicities appeared in commercials aired on a certain city's
television stations, resulting in the data in the accompanying
table.
Ethnicity
African-
American
Asian
Caucasian
Hispanic
Observed Frequency
59
11
332
6
Based on the 2000 Census, the proportion of the U.S. population
falling into each of these four ethnic groups are 0.177 for
African-American, 0.032 for Asian, 0.734 for Caucasian,...

Birds use color to select and avoid certain types of food. The
authors of a particular article studied the pecking behavior of
1-day-old bobwhites. In an area painted white, they inserted four
pins with different colored heads. The color of the pin chosen on
the bird's first peck was noted for each of 34 bobwhites, resulting
in the accompanying table.
Color
First
Peck
Frequency
Blue
15
Green
9
Yellow
7
Red
3
Do these data provide evidence of a color...

USA Today ("Hybrid Car Sales Rose 81% Last Year," April
25, 2005) reported the top five states for sales of hybrid cars in
2004 as California, Virginia, Washington, Florida, and Maryland.
Suppose that each car in a sample of 2004 hybrid car sales is
classified by state where the sale took place. Sales from states
other than the top five were excluded from the sample, resulting in
the accompanying table.
State
Observed
Frequency
California
255
Virginia
56
Washington
37
Florida...

USA Today ("Hybrid Car Sales Rose 81% Last Year," April
25, 2005) reported the top five states for sales of hybrid cars in
2004 as California, Virginia, Washington, Florida, and Maryland.
Suppose that each car in a sample of 2004 hybrid car sales is
classified by state where the sale took place. Sales from states
other than the top five were excluded from the sample, resulting in
the accompanying table.
State
Observed Frequency
California
255
Virginia
56
Washington
37
Florida...

The authors of a paper classified characters who were depicted
smoking in movies released between a certain range of years. The
smoking characters were classified according to sex and whether the
character type was positive, negative, or neutral. The resulting
data is given in the accompanying table.
Assume that it is reasonable to consider this sample of smoking
movie characters as representative of smoking movie characters. Do
the data provide evidence of an association between sex and
character type for...

Suppose that a response can fall into one of k = 5 categories
with probabilities p1, p2, , p5 and that n = 300 responses produced
these category counts.
Category 1 2 3 4 5
Observed Count 46 63 73 52 66
(a) Are the five categories equally likely to occur? How would
you test this hypothesis?
H0: At least one pi is different from 1 5 . Ha: p1 = p2 = p3 =
p4 = p5 = 1...

The authors of a paper concerned about racial stereotypes in
television counted the number of times that characters of different
ethnicities appeared in commercials aired on a certain city's
television stations, resulting in the data in the accompanying
table.
Ethnicity
African-
American
Asian
Caucasian
Hispanic
Observed Frequency
58
12
322
6
Based on the 2000 Census, the proportion of the U.S. population
falling into each of these four ethnic groups are 0.177 for
African-American, 0.032 for Asian, 0.734 for Caucasian,...

A report summarizes a survey of people in two independent random
samples. One sample consisted of 800 young adults (age 19 to 35),
and the other sample consisted of 300 parents of young adults age
19 to 35. The young adults were presented with a variety of
situations (such as getting married or buying a house) and were
asked if they thought that their parents were likely to provide
financial support in that situation. The parents of young adults
were...

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