AP Statistics Variables and Categorical Data Displays Worksheet Solutions (PDF)

A variable is quantitative only if arithmetic on its values makes sense.

1. Categorical (genre names are labels).
2. Quantitative (a count of messages).
3. Categorical: an area code names a region, and adding area codes is meaningless.
4. Quantitative (a measurement).
5. Categorical: sizes are labels for groups.

Answers: categorical, quantitative, categorical, quantitative, categorical.

The individuals are the swimmers: Maya, Jamal, and Sofia.

The variables are grade, best time, and stroke.

• Grade: quantitative (a number that can be compared and averaged).
• Best 50 m time: quantitative.
• Stroke: categorical.

3 individuals, 3 variables: 2 quantitative and 1 categorical.

Divide each count by the total \(n=200\).

1. \(\frac{84}{200}=0.42\), or 42%
2. \(\frac{62}{200}=0.31\), or 31%
3. \(\frac{34}{200}=0.17\), or 17%
4. \(\frac{20}{200}=0.10\), or 10%

Check: \(0.42+0.31+0.17+0.10=1.00\).

Bus 42%, car 31%, walk 17%, bike 10%.

The percents must total 100%, so other = \(100\%-46\%-22\%-18\%=14\%\).

Count: \(0.14\times 250=35\).

14%, which is 35 teens.

(a) \(24+36+12+18=90\) students.

(b) Chocolate has the tallest bar.

(c) \(\frac{36}{90}=0.40\).

90 students; chocolate; 40%.

A full circle is \(360^{\circ}\), so each 1% is \(3.6^{\circ}\).

1. \(35\times 3.6=126^{\circ}\)
2. \(20\times 3.6=72^{\circ}\)
3. \(15\times 3.6=54^{\circ}\)
4. \(10\times 3.6=36^{\circ}\)
5. \(20\times 3.6=72^{\circ}\)

Check: \(126+72+54+36+72=360\).

126, 72, 54, 36, 72 degrees.

Add them: \(0.28+0.35+0.19+0.12=0.94\).

The total must be 1, so these cannot be the complete distribution. There is a missing category of \(1-0.94=0.06\).

No; a category with relative frequency 0.06 is missing.

Tally each response.

Total 20, and the relative frequencies add to 1.

Dog 9, cat 6, fish 3, bird 2.

On this axis, the visible heights are \(92-90=2\) and \(96-90=6\) units.

The Main Street bar looks \(\frac{6}{2}=3\) times as tall, but the real ratio is \(\frac{96}{92}\approx 1.04\).

The bar looks 3 times as tall but the rate is only about 4% higher. A bar graph should start at 0.

Solve \(0.19n=57\).

\(n=\frac{57}{0.19}=300\).

Not soccer: \(300-57=243\).

300 students polled; 243 chose something else.

No. Jersey numbers identify players; they do not measure an amount. A jersey 20 is not twice as much of anything as jersey 10.

Jersey number is categorical, and its mean has no meaning.

1. Either works: each student is in exactly one grade, so the parts make a whole.
2. Bar graph only. The percents add to more than 100%, so a pie chart would be wrong.
3. Either works; the 5 percents add to 100%.

Pie chart is allowed in (a) and (c) only; a bar graph is safe in all three.

Ninth graders: \(\frac{20}{80}=25\%\), \(\frac{44}{80}=55\%\), \(\frac{16}{80}=20\%\).

Twelfth graders: \(\frac{15}{125}=12\%\), \(\frac{35}{125}=28\%\), \(\frac{75}{125}=60\%\).

Twelfth graders: 60% drive compared with 20% of ninth graders.

The groups have different sizes: 80 and 125 students. The counts differ partly because the groups differ in size.

Percents of each group, 20% and 60%, remove the effect of group size.

Use relative frequencies when group sizes differ.

Multiply each percent by 600.

1. \(41\%\times 600=246\)
2. \(33\%\times 600=198\)
3. \(17\%\times 600=102\)
4. \(9\%\times 600=54\)

Music or video: \(41\%+33\%=74\%\), or \(246+198=444\) teens.

246, 198, 102, 54 teens; 74% chose music or video.

Calculators: \(150-54-36-45=15\).

Relative frequencies: \(\frac{54}{150}=0.36\), \(\frac{36}{150}=0.24\), \(\frac{45}{150}=0.30\), \(\frac{15}{150}=0.10\).

Not snacks: \(1-0.30=0.70\).

15 calculators; 70% of sales were not snacks.

Percents: \(35\%\), \(22.5\%\), \(15\%\), \(27.5\%\).

Sample answer: Pizza was the most popular lunch, chosen by 35% of the 120 students, and salad was the least popular at 15%. Tacos (27.5%) and burgers (22.5%) were in the middle.

Other: \(100\%-38\%-29\%-21\%=12\%\).

Brand A: \(0.38\times 480=182.4\), about 182. Brand C: \(0.21\times 480=100.8\), about 101.

Difference: \((0.38-0.21)\times 480=0.17\times 480=81.6\), about 82 owners.

About 82 more owners use brand A; 12% use other brands.

Ms. Lee: \(\frac{14}{28}=0.50\). Mr. Cruz: \(\frac{18}{40}=0.45\).

The class sizes differ, so percents are the fair comparison.

Yes, 50% of Ms. Lee’s class compared with 45% of Mr. Cruz’s class prefer fall, but the difference is small.

Adults: \(2400-1080-720=600\).

New teens: \(720\times 1.20=864\).

New total: \(1080+864+600=2544\).

1. \(\frac{1080}{2544}\approx 0.4245\)
2. \(\frac{864}{2544}\approx 0.3396\)
3. \(\frac{600}{2544}\approx 0.2358\)

Children 42.5%, teens 34.0%, adults 23.6% (they add to 100.1% because of rounding).