A travel agency claims that the reasons people choose their vacation destinations are distributed as follows: $35\%$ for relaxation, $25\%$ for adventure, $20\%$ for cultural experiences, and $20\%$ for family visits. In a survey of $900$ travelers, $310$ chose relaxation, $220$ chose adventure, $180$ chose cultural experiences, and $190$ chose family visits. At $\alpha =0.05$, test whether the agency's claim about the distribution is supported by the data.

A researcher collected data on the drying times (in minutes) of four types of wall paint: matte, satin, semi-gloss, and gloss. The data were collected under controlled temperature and humidity conditions. The drying times (in minutes) are listed below:

Matte: $38,41,43,39,40$

Satin: $44,47,45,46,43$

Semi-gloss: $50,49,51,48,52$

Gloss: $42,40,41,43,44$

At the $0.05$ significance level, use the Kruskal-Wallis test to determine whether there is a significant difference in the median drying times among the four paint types.

A health researcher claims that people's favorite types of exercise differ from the preferences suggested by previous national fitness survey data. According to the national survey, the distribution of exercise preferences is as follows:

Running: $35\%$

Weightlifting: $25\%$

Yoga/Pilates: $20\%$

Other: $20\%$

To test this claim, the researcher randomly selects $500$ adults and records their favorite type of exercise. The results are shown in the table below.

Find the critical value and identify the rejection region if $\alpha =0.05$.

A hospital administrator claims that emergency room (ER) visits occur equally across all days of the week. To test this, a random sample of $840$ ER visits from the past month is analyzed. The number of visits per day is recorded below:

Test the administrator’s claim at the $\alpha =0.05$ significance level.

A streaming service claims that the distribution of user preferences for movie genres follows this national pattern:

Action: $30\%$

Comedy: $25\%$

Drama: $20\%$

Horror: $15\%$

Science Fiction: $10\%$

A survey of $400$ randomly selected users from the platform gives the following results:

At $\alpha =0.05$, test the claim that the streaming service users' preferences match the national distribution.

Fill in the blanks.

In order to perform a goodness-of-fit test, the two main requirements are that the data must be in the form of ____________________ and each expected frequency should be at least ___.

Determine the degrees of freedom for a ${\chi }^2$ test of goodness of fit, given that there are $4$ beverage categories (Coffee, Tea, Juice, and Water) in a survey with $200$ responses.

Fill in the blanks.

A ________________ is used to assess whether there is a significant association between two categorical variables.

A forensic accountant is examining $650$ sales invoices from a company suspected of financial irregularities. She records the leading digits of each invoice amount and compares the observed distribution to Benford’s Law, which describes the expected frequency of leading digits in naturally occurring datasets.

The observed frequencies and Benford’s expected percentages are shown below:

At the $0.01$ significance level, test whether the observed digit distribution follows Benford’s Law.

A health researcher claims that people's favorite types of exercise differ from the preferences suggested by previous national fitness survey data. According to the national survey, the distribution of exercise preferences is as follows:

Running: $35\%$

Weightlifting: $25\%$

Yoga/Pilates: $20\%$

Other: $20\%$

To test this claim, the researcher randomly selects $500$ adults and records their favorite type of exercise. The results are shown in the table below.

Find the chi-square test statistic if $\alpha =0.05$.