Random sampling in 7th grade math

The correct answer is the statement about equal chance. Using a random number generator on the full membership list gives every member a chance to be selected, so the sample is less likely to favor one group and can support an inference about this library.

1. 1The librarian used a random number generator to select 40 members from the full library membership list. This means every member of the library was part of the pool from which the sample was drawn.
2. 2Because every library member had an equal chance to be selected, no group of members was intentionally left out or favored. The sample is not hand-picked, so it can tend to reflect all library members.
3. 3The correct explanation says every member had an equal chance and no group was intentionally favored, so the sample can tend to represent all library members. Random sampling does not guarantee the exact same percentage in the sample, does not make people answer more honestly, and does not describe every library.

Common mistake. A common mistake is thinking a random sample will have exactly the same percentage as the whole group. Random samples tend to reflect the population but do not have to match it exactly.

The idea underneath. Random sampling works because equal chance of selection reduces favoritism, so the sample tends to mirror the population instead of a hand-picked subgroup.

The correct matches are: about 7 books → Reasonable claim; about 11 books → Estimate is too high; about 3 books → Estimate is too low; exactly 7 books → Too certain from one sample.

1. 1Look at the line plot. The 9 students read 4, 6, 6, 7, 7, 7, 8, 8, and 10 books. The fewest is 4 and the most is 10.
2. 2Find the sample mean. Add all 9 values and divide by 9. The sample mean is 7 books.
3. 3The claim 'The population average is about 7 books.' equals the sample mean, so it is a Reasonable claim. The students were chosen at random, so the sample mean is a good estimate of the school's average.
4. 4The claim 'The population average is about 11 books.' is too high. Every student in the sample read 10 books or fewer, which is lower than 11. A random sample from a school that averages 11 books would likely include some students who read 11 or more.
5. 5The claim 'The population average is about 3 books.' is too low. Every student in the sample read 4 books or more, which is higher than 3. A random sample from a school that averages 3 books would likely include some students who read 3 or fewer.
6. 6The claim 'The population average is exactly 7 books.' is Too certain from one sample. A different random sample would likely give a different mean. One sample can support an estimate near 7, but it cannot prove the exact average.

Common mistake. A common mistake is marking 'exactly 7 books' as reasonable because 7 is the sample mean. One random sample supports an estimate near 7, but a different sample would likely give a different mean, so it cannot prove the exact average.

The idea underneath. Each student at the school had the same chance of being chosen, so a random sample tends to look like the whole school. That is why the sample mean is a good estimate of the school's average. A random sample also tends to include students both above and below the school's average, so a claim that is higher or lower than every student in the sample does not fit. A different random sample would likely give a different mean, so one sample can support an estimate but cannot prove an exact average.

The sample is the 20 customers shopping on Friday afternoon because they are the group she actually surveyed. The population is all customers at Maple Grocery, and the other two phrases name the topic and the researcher.

1. 1The population is the larger group the store manager wants to learn about. She wants to learn about all customers at Maple Grocery, so that phrase goes in the Population bin.
2. 2The sample is the smaller group she actually surveyed. She surveyed the 20 customers shopping on Friday afternoon, so that phrase goes in the Sample bin.
3. 3The other two phrases do not name a group of people who were studied. "Favorite snacks" is the topic she wants to learn about, and "the store manager" is the person running the study. Both go in the Neither bin.

Common mistake. A common mistake is placing 'all customers at Maple Grocery' in the Sample bin because it names a group of customers. That phrase is the population — only the 20 Friday afternoon shoppers were actually surveyed.

The idea underneath. The sample is the group that data was actually collected from. The population is the larger group the study is about. The topic being measured and the person collecting the data are neither.

15 estimates are in the inclusive interval from 58% to 60%, and this is a likely interval for the population percent because it sits at the center of the sample estimates.

1. 1Look at the inclusive interval from 58% to 60%. Because the interval is inclusive, dots placed exactly at 58% and at 60% both count.
2. 2Count the dots in each stack inside the interval: 58% has 4 dots, 59% has 5 dots, and 60% has 6 dots.
3. 3Add the counts across the three stacks.
4. 415 of the 31 estimates (about 48%) fall in this interval, which sits at the center of the distribution. Because so many random samples cluster here, the population percent is likely to be between 58% and 60%.

Common mistake. A common mistake is to count the 60% stack as 5 instead of 6, giving a total of 14, or to add one dot from the 61% stack to get 16. Another mistake is to call a densely populated central interval unlikely, even though the center of the distribution is where the population percent most likely falls.

The idea underneath. Repeated random sample estimates tend to cluster around the population value, so intervals with many dots near the center are more plausible for the population than intervals in the tails.