Suppose it is known that 20 % of students at a certain...

Suppose it is known that $20 \%$ of students at a certain college participate in a textbook recycling program each semester. a. If a random sample of 50 students is selected, do we expect that exactly $20 \%$ of the sample participates in the textbook recycling program? Why or why not? b. Suppose we take a sample of 500 students and find the sample proportion participating in the recycling program. Which sample proportion do you think is more likely to be closer to $20 \%$ : the proportion from a sample size of 50 or the proportion from a sample size of $500 ?$ Explain your reasoning.

Suppose it is known that $20 \%$ of students at a certain college participate in a textbook recycling program each semester.
a. If a random sample of 50 students is selected, do we expect that exactly $20 \%$ of the sample participates in the textbook recycling program? Why or why not?
b. Suppose we take a sample of 500 students and find the sample proportion participating in the recycling program. Which sample proportion do you think is more likely to be closer to $20 \%$ : the proportion from a sample size of 50 or the proportion from a sample size of $500 ?$ Explain
your reasoning.

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Key Concepts

Sampling Variability

Sampling variability refers to the natural fluctuations that occur in sample statistics, such as the sample proportion, because of the randomness involved in selecting a subset from a population. This means that even if the true population proportion is 20%, any given sample might show a proportion that slightly deviates from 20% due to chance.

Law of Large Numbers

The Law of Large Numbers states that as the sample size increases, the sample estimate becomes more likely to be close to the true population parameter. In other words, larger samples are more stable and reliable, ensuring that the sample proportion will tend to converge to the actual rate of 20% in the population.

Sample Size and Standard Error

The standard error is a measure of how much a sample statistic, like a proportion, is expected to vary from the true population parameter, and it decreases as the sample size increases. A larger sample size reduces this variability, making it more likely that the measured proportion will be very close to the true value of 20%.

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