Middle School Math Quiz: Draw Inferences From Random Samples

What this quiz covers

This quiz focuses on Draw Inferences From Random Samples, giving you a quick way to practice the rules, question types, and explanations that matter most for Middle School Math.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

Question 1

1. The teacher's approach, because larger samples always provide more accurate estimates of population parameters
2. The teacher's approach, because it eliminates the confusion caused by getting different results from multiple samples
3. Both approaches are equally good since they survey the same total number of students overall
4. The nurse's approach, because multiple samples reveal how much estimates can vary and improve reliability (correct answer)

Question 2

1. Martinez will definitely win because she led in two out of three surveys conducted
2. The election is too close to call reliably, given the variation and small margins in the sample data (correct answer)
3. Chen will likely win because Survey 2 showed the largest margin of victory for any candidate
4. Martinez will probably win because her average support across all surveys exceeds 50% of responses

Question 3

1. Predict exactly 14.96 points per game by averaging all sample means and multiplying by appropriate factors
2. Predict between 14.0 and 15.5 points per game using only the three most consistent sample results
3. Predict 16.1 points per game since this was the highest average observed in the samples
4. Predict the season average will fall between 13.5 and 16.5 points per game based on sample variation (correct answer)

Question 4

1. She should discard the 21.1 ft result because it differs too much from the other samples
2. She should expand her estimated range to account for this additional variation in the data (correct answer)
3. She should conclude that oak trees are getting taller, since the final sample showed the highest average
4. She should repeat the final sample because 21.1 ft is probably a measurement error

Question 5

1. Customer satisfaction improved over time, since the later samples had higher average ratings than earlier ones
2. The true average satisfaction is approximately 3.9, calculated as the mean of all five sample averages
3. Customer satisfaction likely ranges between 3.4 and 4.3, reflecting the uncertainty shown by sample variation (correct answer)
4. The sampling method is flawed because the results vary too much to provide useful information

Question 6

1. The population mean is exactly 3.5 books because it is between the other two sample means.
2. The sample means vary by about 0.6 books (from 3.2 to 3.8), so the population mean is likely around 3.5 books, but not exactly. (correct answer)
3. The population mean must be 15 books because each sample had 15 students.
4. Because the sample means are different, it is impossible to estimate the population mean.

Question 7

1. The estimates vary by about 3 minutes (22 to 25), so the population mean is likely around 23–24 minutes. (correct answer)
2. The estimates should be identical; since they are different, the sampling must be wrong.
3. The population mean must be exactly 25 minutes because that is the largest sample mean.
4. The estimates vary by 30 minutes because each sample had 30 students, so the population mean is 30 minutes.

Question 8

1. The population mean must be exactly 52 minutes because Sample 1 is larger than Sample 2.
2. The population mean must be 40 minutes because 40 students were sampled.
3. The estimates differ by 5 minutes, showing sampling variability; the population mean is likely somewhere around 50 minutes. (correct answer)
4. Because the means are different, at least one sample is not random.

Question 9

1. About $\frac{39}{60}\approx 35\%$ of all 7th graders prefer a longer lunch.
2. About $\frac{39}{60}\approx 65\%$ of all 7th graders prefer a longer lunch. (correct answer)
3. Exactly $65\%$ of all 7th graders prefer a longer lunch with no possible error.
4. Exactly 39 seventh graders in the whole grade prefer a longer lunch.

Question 10

1. Exactly 46 students in the whole school prefer pretzels.
2. Popcorn is definitely more popular because the sample is not a census.
3. About $\frac{46}{80}=57.5\%$ of students prefer pretzels, so pretzels are likely more popular than popcorn. (correct answer)
4. Exactly 57.5% of the whole school prefers pretzels with no uncertainty.

Question 11

1. All rocks in the bucket have mass 180 grams.
2. The population mean mass is 15 grams because 15 rocks were selected.
3. The population mean mass of rocks in the bucket is likely close to 180 grams. (correct answer)
4. No estimate can be made from a random sample.

Question 12

1. About $\frac{39}{60}=65\%$ of 7th graders prefer a longer lunch. (correct answer)
2. Exactly 60% of 7th graders prefer a longer lunch.
3. Exactly 39% of 7th graders prefer a longer lunch.
4. About $\frac{39}{60}=35\%$ of 7th graders prefer a longer lunch.

Question 13

1. The population mean is exactly 3.5 books because it is between the other two sample means.
2. The sample means vary by about 0.6 books (from 3.2 to 3.8), so the population mean is likely around 3.5 books, but not exactly. (correct answer)
3. The population mean must be 15 books because each sample had 15 students.
4. Because the sample means are different, it is impossible to estimate the population mean.

Question 14

1. The two different sample means show sampling variability; the population mean is likely around 42 grams. (correct answer)
2. No estimate of the population mean can be made from samples.
3. The population mean mass is exactly 42.6 grams, and the 41.9 grams must be an error.
4. The population mean mass must be 30 grams because 30 rocks were sampled.

Question 15

1. The estimates are 28% and 31%, so they differ by 3 percentage points.
2. The population proportion must be exactly 62% because Sample 2 is larger.
3. The estimates are $\frac{28}{50}=56\%$ and $\frac{31}{50}=62\%$, so they differ by about 6 percentage points. (correct answer)
4. Because the estimates are different, at least one sample was not random.

Question 16

1. About $\frac{46}{80}=57.5\%$ of students will vote Yes, so Yes will probably get a little more than half the votes. (correct answer)
2. Exactly 57.5% of students will vote Yes, with no chance of error.
3. No prediction can be made because a sample is not useful.
4. Exactly 46% of students will vote Yes.

Question 17

1. About 20 seconds
2. About 0.7 seconds
3. About 78.4 seconds
4. About 1.7 seconds (correct answer)

Question 18

1. The true population percentage is exactly 60% because the middle sample is 48/80.
2. The estimates vary by about 50 percentage points, so the sampling method failed.
3. The estimates are 46/80, 50/80, and 48/80 (about 58%-63%), varying by about 5 percentage points, so the true value is likely around 60%. (correct answer)
4. No conclusion can be drawn unless every student in the school votes.

Question 19

1. About $\pm 0.2$ letters, since the means range from 4.1 to 4.4. (correct answer)
2. About $\pm 2.0$ letters, because 20 words were sampled.
3. 0 letters, because a random sample gives the exact population mean.
4. About $\pm 0.03$ letters, because the range is 0.03.

Question 20

1. A single sample mean could reasonably be off by around 5 cm or so, since the sample means vary by about 9 cm from lowest to highest. (correct answer)
2. The garden must have exactly 10 sunflowers because each sample had 10.
3. A single sample mean must be off by exactly 9 cm.
4. A single sample mean will always equal the population mean exactly.