Representing a Categorical Variable with Tables - AP Statistics

1. A frequency table is used to summarize categorical data. Which of the following does a frequency table directly display?
   Answer: The number of observations falling into each category. A frequency table shows the count, or number, of observations for each category of a categorical variable. A relative frequency table shows the proportion or percentage of observations in each category.
2. In a survey of 400 office workers, the relative frequency of those who prefer to work from home was found to be 0.65. How many office workers in the sample prefer to work from home?
   Answer: 260 The frequency (count) is found by multiplying the total number of observations by the relative frequency. The calculation is $400 \times 0.65 = 260$. The other choices are incorrect calculations.
3. A technology teacher wants to summarize which device students most often use to complete homework. A random sample of 90 students in the school is selected, and each student reports the primary device used (a categorical variable). The distribution is shown below. Which statement is supported by the data?

   Primary homework device	Frequency	Relative frequency
   Laptop	39	0.433
   Tablet	18	0.200
   Phone	24	0.267
   Desktop	9	0.100

   Answer: Laptops are the most common primary device in the sample, at about 43.3%. This problem focuses on the skill of representing a categorical variable with tables, interpreting frequency and relative frequency for primary homework devices in 90 students. Choice C is supported, with laptops at 0.433 or about 43.3%, the most common, surpassing phones at 26.7%. This is backed by 39/90 ≈ 0.433, the highest. A typical distractor is choice E, stating the frequency for phone is 0.267 students, confusing relative frequency with the actual count of 24. Choice D wrongly adds desktops and tablets to 63.3%, but it's 30%. For interpreting categorical tables, use relative frequencies to assess prevalence and add them for combinations, ensuring you reference the correct column to avoid mixing counts and proportions.
4. A table with three categories gives 26%, 38%, 39%; each was rounded to nearest percent. Their sum is 103%. What is true?
   Answer: Rounded values sum too high Each rounded value can be off by no more than half a percentage point, so three rounded values can deviate from 100 by at most 1.5 points, putting the total between 98.5 and 101.5. A sum of 103 is outside that range, so the rounded values sum too high. You might be tempted to rescale by 103, but that only forces a new total of 100 instead of explaining why the original rounded entries are inconsistent.
5. Car 52%, bus 23%, bike 18%, walk 7%. A cumulative table gives 52, 75, 93, 100. The main flaw is that...
   Answer: Category order is arbitrary A cumulative table only makes sense when the categories have a meaningful order; car, bus, bike, and walk do not, so their sequence is arbitrary and the cumulative totals are meaningless. The last entry of 100 is fine because the table uses percentages, not decimal proportions. The tempting wrong answer is that the last entry should be 1.00, but that only applies if the table used decimals like 0.52 instead of 52.
6. A survey of 200 customers at a coffee shop found that 110 ordered regular coffee, 50 ordered tea, and 40 ordered a specialty drink. Which statement is a correct interpretation of the relative frequency for tea drinkers?
   Answer: The proportion of customers in the sample who ordered tea was 0.25. The relative frequency of tea drinkers is the count of tea drinkers divided by the total number of customers, which is $50 / 200 = 0.25$. This value represents the proportion within the sample. Choices A and B incorrectly swap the definitions of frequency and relative frequency. Choice D makes an incorrect comparison.
7. What proportion of patients in the sample have Type O blood?
   Answer: 0.30 First, find the frequency of Type O blood by subtracting the other frequencies from the total: $50 - (20 + 11 + 4) = 50 - 35 = 15$. Then, calculate the proportion (relative frequency) for Type O: $15 / 50 = 0.30$.
8. A relative frequency table for the preferred social media platform of a group of teenagers shows that 0.35 prefer Platform A, 0.25 prefer Platform B, and 0.20 prefer Platform C. If the remaining teenagers in the group prefer Platform D, what is the relative frequency for Platform D?
   Answer: 0.20 The sum of all relative frequencies in a distribution must be 1. The sum of the given relative frequencies is $0.35 + 0.25 + 0.20 = 0.80$. Therefore, the relative frequency for Platform D must be $1 - 0.80 = 0.20$.
9. A public library is evaluating how adult patrons prefer to access books. From the population of adult patrons who used the library this month, the library selects a random sample of 80 patrons and records each patron's preferred format (a categorical variable). The distribution is shown below. Which statement is supported by the data?

   Preferred book format	Frequency	Relative frequency
   Print	44	0.550
   E-book	20	0.250
   Audiobook	12	0.150
   Other	4	0.050

   Answer: Print is the most common preferred format in the sample, at about 55%. The skill here is representing a categorical variable with tables, interpreting frequency and relative frequency for preferred book formats in a sample of 80 patrons. Choice D is backed by the data, with print at 0.550 or 55%, the highest proportion, outranking e-books at 25%. Evidence comes from 44/80 = 0.55, establishing it as the most common. A common distractor is choice B, claiming audiobooks at about 12%, but it's 0.150 or 15%, likely a misreading of the table. Choice E says other at 4%, but it's 0.050 or 5%, a slight but incorrect approximation. A useful mini-lesson on categorical tables is to rely on relative frequencies for percentage claims and compare them to identify the least or most preferred options, ensuring approximations align closely with exact values.
10. A relative frequency table is given for the political affiliation of 500 voters in a district: Party A: 0.44, Party B: 0.38, Party C: 0.12, Independent: 0.06. A commentator states that Parties A and B together represent an overwhelming majority of voters. What is the combined frequency of voters for Party A and Party B?
    Answer: 410 First, find the combined relative frequency for Party A and Party B: $0.44 + 0.38 = 0.82$. Then, convert this proportion back to a frequency by multiplying by the total number of voters: $0.82 \times 500 = 410$.
11. A city transportation office is studying commuting habits of adults who live within city limits. A random sample of 200 adults is asked to report their primary commute mode (a categorical variable). The distribution is shown below. Which statement is supported by the data?

    Primary commute mode	Frequency	Relative frequency
    Drive alone	98	0.490
    Carpool	34	0.170
    Public transit	46	0.230
    Bike/Walk	22	0.110

    Answer: About 49% of sampled adults reported driving alone, which is the largest category. The skill tested here involves representing a categorical variable with tables, focusing on interpreting frequencies and relative frequencies for commute modes in a sample of 200 adults. Choice C is supported as the relative frequency for driving alone is 0.490, or about 49%, and it is the largest category, exceeding public transit at 23% and others. This is evidenced by the frequency of 98, which is 98/200 = 0.49, confirming it's the mode. A frequent distractor is choice E, claiming the frequency for bike/walk is 0.110 adults, but 0.110 is the relative frequency, while the actual frequency is 22, mixing up the two columns. Choice D incorrectly adds carpool and bike/walk to over half (0.170 + 0.110 = 0.280), which is less than 0.5. A transferable mini-lesson for categorical tables is to always verify proportions by dividing frequencies by the total and compare them to assess dominance or combinations, avoiding assumptions about aggregates without calculation.
12. A dataset contains observations on the eye color of 80 individuals. If a frequency table is created for this data, which of the following must be true?
    Answer: The sum of the frequencies will be equal to 80. The sum of the frequencies (counts) in a frequency table must equal the total number of observations in the dataset, which is 80 in this case. The other statements are incorrect definitions or properties of frequency tables.
13. A high school guidance counselor surveyed 250 graduating seniors about their post-graduation plans. The results showed that 120 plan to attend a four-year college, 75 plan to attend a two-year college, 30 plan to enter the workforce, and 25 are undecided. What is the relative frequency of seniors who plan to attend a four-year college?
    Answer: 0.48 The relative frequency is the count for the category divided by the total number of observations. In this case, it is $\frac{120}{250} = 0.48$. Choice B is the relative frequency for entering the workforce ($\frac{30}{250} = 0.12$, a calculation error could lead to 0.30). Choice D is the frequency, not the relative frequency.
14. Which of the following represents the relative frequency table for the pollution levels of the 150 rivers?
    Answer: Low: 0.42, Medium: 0.30, High: 0.28 First, find the frequency for High: $150 - 63 - 45 = 42$. Then, calculate relative frequencies: Low is $63/150 = 0.42$, Medium is $45/150 = 0.30$, and High is $42/150 = 0.28$. Choice A correctly lists these relative frequencies. Choice B lists frequencies. Choices C and D use incorrect calculations.
15. A library tracks the genres of books checked out by patrons. A relative frequency table shows the following: Fiction: 0.52, Non-Fiction: 0.28, Young Adult: 0.15. The rest are Children's books. If 400 books were checked out in total, what is the frequency of Children's books?
    Answer: 20 First, find the relative frequency of Children's books. The sum of relative frequencies must be 1. $1 - (0.52 + 0.28 + 0.15) = 1 - 0.95 = 0.05$. Then, find the frequency by multiplying this relative frequency by the total number of books: $0.05 \times 400 = 20$.
16. To convert a relative frequency table back to a frequency table, what additional piece of information is required?
    Answer: The total number of observations. A relative frequency is a proportion. To find the original count (frequency), you must multiply the proportion by the total number of observations (the sample size). Without the total number of observations, the original frequencies cannot be recovered.
17. Which of the following values represents the relative frequency of homeowners who use natural gas?
    Answer: 0.55 The relative frequency is the count for a category divided by the total number of observations. For natural gas, this is $66 / 120 = 0.55$. Choice A is the frequency. Choices B and D are incorrect calculations.
18. A survey of 1,200 residents: 480 rent, 360 own, 240 other. A table says rent 0.40, own 0.30, other 0.30. What is wrong?
    Answer: Other should be 0.20, not 0.30 Among 1,200 residents, each proportion is count divided by 1,200: rent 480/1,200 = 0.40, own 360/1,200 = 0.30, other 240/1,200 = 0.20. So the table's 0.30 for other is wrong. The tempting mistake is thinking rent is 0.48, which comes from dividing 480 by 1,000 instead of 1,200.
19. What is the relative frequency of votes for Candidate A, expressed as a percentage?
    Answer: 45% First, find the total number of votes: $450 + 300 + 250 = 1000$. The relative frequency for Candidate A is $450/1000 = 0.45$. To express this as a percentage, multiply by 100, which gives 45%.
20. A university dining services team is researching beverage choices among students who eat lunch on campus. A random sample of 150 students is asked to choose their most common lunch beverage (a categorical variable). The results are summarized below. Which statement is supported by the data?

    Lunch beverage	Frequency	Relative frequency
    Water	63	0.420
    Soda	36	0.240
    Coffee/Tea	30	0.200
    Juice	21	0.140

    Answer: Water is the most common beverage in the sample, at about 42%. This problem evaluates the ability to represent a categorical variable with tables by analyzing frequency and relative frequency for lunch beverage choices among 150 students. The data supports choice C since water has a relative frequency of 0.420, or 42%, and it's the highest, surpassing soda at 24%. Supporting evidence includes the frequency of 63, where 63/150 = 0.42, clearly the mode. A typical distractor is choice A, stating juice is about 21%, but 0.140 is 14%, possibly mistaking the frequency 21 for a percentage. Choice E errs by saying the relative frequency for coffee/tea is 30, confusing it with the frequency count. For interpreting categorical tables, remember to differentiate counts from proportions and use relative frequencies to determine popularity, calculating sums for combined categories to check claims like majorities.

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