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14.3 The authors of the paper "Is It Really About Me? Message Content in Social Awareness Streams" (Computer Supported Cooperative Work 2010) studied a random sample of 350 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: The accompanying table gives the observed counts for the five categories (approximate values from a graph in the paper). 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 \(\alpha=0.05 .\) (Hint: See Example 14.3.)

Packages of mixed nuts made by a certain company contain four types of nuts. The percentages of nuts of Types \(1,2,3,\) and 4 are advertised to be \(40 \%, 30 \%, 20 \%,\) and \(10 \%,\) respectively. A random sample of nuts is selected, and each one is categorized by type. a. If the sample size is \(n=200\) and the resulting test statistic value is \(X^{2}=19.0,\) what conclusion would be appropriate for a significance level of \(0.001 ?\) b. If the random sample had consisted of only 40 nuts, would you use the chi- square goodness-of-fit test? Explain your reasoning.

What is the approximate \(P\) -value for the following values of \(X^{2}\) and \(\mathrm{df}\) ? a. \(X^{2}=34.52, \mathrm{df}=13\) b. \(X^{2}=39.25, \mathrm{df}=16\) c. \(X^{2}=26.00, \mathrm{df}=19\)

The authors of the paper "External Factors and the Incidence of Severe Trauma: Time, Date, Season and Moon" (Injury [2014]: S93-S99) classified admissions to hospitals in Germany according to season. They wondered if severe trauma injuries were more common in some seasons than others. For purposes of this exercise, assume that there were 1200 trauma cases in the sample and that the sample is representative of severe trauma injuries in Germany. The data in the accompanying table are consistent with summary quantities given in the paper. Do these data support the theory that the proportion of severe trauma cases is not the same for the four seasons? Test the relevant hypotheses using a significance level of 0.05 .

Give an example of a situation where it would be appropriate to use a chi- square test of homogeneity. Describe the populations that would be sampled and the variable that would be recorded.

Explain the difference between situations that would lead to a chi-square goodness-of-fit test and those that would lead to a chi-square test of homogeneity.

Explain the difference between situations that would lead to a chi-square test for homogeneity and those that would lead to a chi-square test for independence.

Does viewing angle affect a person's ability to tell the difference between a female nose and a male nose? This important (?) research question was examined in the article "You Can Tell by the Nose: Judging Sex from an Isolated Facial Feature" (Perception [1995]: 969-973). Eight Caucasian males and eight Caucasian females posed for nose photos. The article states that none of the volunteers wore nose studs or had prominent nasal hair. Each person placed a black Lycra tube over his or her head in such a way that only the nose protruded through a hole in the material. Photos were then taken from three different angles: front view, three-quarter view, and profile. These photos were shown to a sample of undergraduate students. Each student in the sample was shown one of the nose photos and asked whether it was a photo of a male or a female. The response was classified as either correct or incorrect. The accompanying table was constructed using summary values reported in the article. Is there evidence that the proportion of correct sex identifications differs for the three different nose views?

The accompanying data on degree of spirituality for a random sample of natural scientists and a random sample of social scientists working at research universities appeared in the paper "Conflict Between Religion and Science Among Academic Scientists" (Journal for the Scientific Study of Religion [2009]: 276-292). Is there evidence that the spirituality category proportions are not the same for natural and social scientists? Test the relevant hypotheses using a significance level of \(\alpha=0.01\)