Random Sampling and Comparing Populations: math practice, Grade 7

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Math practice Grade 7 : Random Sampling and Comparing Populations — Zyro the alien explorer of Planète Maths

21 problems graded by difficulty (★ easy, ★★ medium, ★★★ hard). Try first, without looking at the solutions!

1 Population or sample? ★★★

For each situation, say whether the group described is the population or a sample.

All 420 students at Hillcrest Middle School.
The 35 students whose names are drawn to fill out a lunch survey.
Every tomato plant in a greenhouse.
Ten tomato plants picked at random from that greenhouse.

2 Fair or biased? ★★★

The student council wants to know which field trip all 300 seventh graders prefer. Which methods give a random, representative sample?

Number every seventh grader and pick 30 numbers with a random number generator.
Ask the first 30 students who arrive on the early bus.
Ask the 30 members of the drama club.
Put all 300 names in a bin, mix them, and draw 30 names.

3 Mean, median, and range ★★★

A team of six runners finished a practice mile in these numbers of minutes: 8, 12, 9, 11, 10, 10. Find the mean, the median, and the range.

4 Favorite sport ★★★

A random sample of 40 students is chosen from the 1,200 students at a school. Of the 40, 24 say volleyball is their favorite sport. Estimate how many students in the whole school favor volleyball.

5 Reading a dot plot ★★★

The dot plot shows the number of books ten students read last summer.

Find the mode, the median, the range, and the mean.

6 First MAD ★★★

Find the mean absolute deviation of 4, 6, 8, 10, 12.

7 True or false? ★★★

Say whether each statement is true or false, and explain.

Larger random samples tend to give more reliable estimates.
A sample of 3 people is just as reliable as a sample of 300 people.
Surveying only the fans at a basketball game gives a good estimate of the favorite sport of everyone in town.

8 Counting maple trees ★★★

A park has 450 trees. A ranger examines a random sample of 30 trees and finds that 12 are maples. Estimate the number of maples in the park, and explain why your answer is only an estimate.

9 Two bakeries ★★★

The weights, in ounces, of five loaves from each bakery are: Bakery A: 15, 16, 16, 16, 17. Bakery B: 14, 15, 16, 17, 18. Find the mean and the MAD for each bakery. Which bakery makes loaves of more consistent weight?

10 Five-number summary ★★★

Ten students’ times, in minutes, to finish a puzzle were 3, 5, 6, 8, 9, 11, 12, 14, 15, 17. Find the minimum, \(Q_1\), median, \(Q_3\), maximum, and the IQR.

11 Small sample, big sample ★★★

Lena and Marcus each estimate the percent of students who walk to school. Lena asks 8 randomly chosen students and 5 walk. Marcus asks 80 randomly chosen students and 36 walk. Find each sample percent. Whose estimate is likely to be more reliable, and why?

12 Water bills ★★★

A random sample of 12 homes had these monthly water bills, in dollars: 42, 38, 45, 50, 41, 47, 39, 44, 46, 43, 40, 41. (a) Estimate the mean monthly bill for all homes in the town. (b) The town has 2,000 homes. Estimate the total amount of all their bills in one month.

13 Gap measured in MADs ★★★

Two classes took the same test. Class 1 has mean 78 and Class 2 has mean 86. Both classes have a MAD of 4. By how many MADs do the means differ? Is the difference big compared with the spread?

14 Comparing two box plots ★★★

The box plots show daily steps, in thousands, for two groups of people.

Which group has the greater median? By how much?
Find the IQR of each group.
Is the difference of medians larger or smaller than each IQR?

15 Spotting and fixing bias ★★★

A principal wants to know whether the 600 students favor a later start time. Plan A: survey only students who stay after school for clubs. Plan B: post an online survey and count whoever answers. For each plan, explain why the result could be biased. Then describe a better plan.

16 Same mean, different spread ★★★

Two basketball players scored these points in 8 games. Player A: 14, 16, 18, 18, 20, 20, 22, 24. Player B: 8, 12, 16, 19, 21, 23, 26, 27. Find the mean, median, and MAD of each player, and compare.

17 Defective bulbs ★★★

A quality checker takes six random samples of 50 bulbs from a factory. The percents of defective bulbs are 4%, 6%, 2%, 8%, 6%, 4%. The factory makes 20,000 bulbs in a day. (a) Estimate the number of defective bulbs using the mean of the sample percents. (b) Use the smallest and largest sample percents to give a range of reasonable values. (c) Why can’t the checker be sure?

18 Missing score ★★★

Four test scores are 78, 85, 90, and 80. A fifth score is added so that the mean of all five is 82. (a) Find the fifth score. (b) Find the MAD of the five scores.

19 Tagging fish ★★★

To estimate the number of fish in a lake, a biologist catches and tags 40 fish and releases them. Later she catches a random sample of 30 fish and finds that 6 have tags. Estimate the number of fish in the lake, assuming the tagged fish have mixed in completely.

20 Does fertilizer help? ★★★

Heights, in cm, of ten seedlings after three weeks. No fertilizer: 12, 14, 15, 15, 16, 16, 17, 18, 18, 19. With fertilizer: 15, 17, 18, 18, 19, 19, 20, 21, 21, 22. Find the mean and MAD of each set. Express the difference of the means as a multiple of the MAD, and describe the overlap.

21 Two cities in July ★★★

Daily high temperatures (°F) in July have these five-number summaries. City A: min 48, \(Q_1\) 55, median 62, \(Q_3\) 70, max 80. City B: min 55, \(Q_1\) 60, median 64, \(Q_3\) 68, max 75. (a) Find the range and IQR of each city. (b) Which city has steadier temperatures? (c) Convert each median to °C using \(C=\dfrac{5}{9}(F-32)\), rounded to the nearest tenth.