Question

A student government states that \(80 \%\) of all students favor an increase in student fees to subsidize a new recreational area. A random sample of \(n=25\) students produced 15 in favor of increased fees. What is the probability that 15 or fewer in the sample would favor the issue if student government is correct? Do the data support the student government's assertion, or does it appear that the percentage favoring an increase in fees is less than \(80 \% ?\)

Short answer

Use the binomial distribution to answer the question. Answer: We found that the probability of having 15 or fewer students in favor of increased fees in a sample of 25 students under the student government's claim is P(X ≤ 15) which can be calculated using cumulative binomial probability function. After comparing the obtained p_value with the significance level (0.05), we can conclude whether the data supports the student government's claim or suggests that the percentage of students favoring an increase in fees is less than 80%.

Step-by-step solution

Identify the binomial distribution parameters

In this problem, we have a binomial distribution with the sample size (n) equal to 25, the probability of success (p) equal to 80% or 0.80, and the number of successful outcomes (x) equal to 15 or fewer.

Calculate the cumulative binomial probability

We need to find the probability of getting 15 or fewer students favoring the issue within the sample, so we will calculate the cumulative probability function (CDF) of the binomial distribution: \(P(X \le 15) = \sum_{k=0}^{15} {25\choose k}(0.80)^k(1-0.80)^{25-k}\) We can use a binomial calculator or statistical software to find the cumulative probability.

Analyze the calculated probability

Suppose the cumulative probability obtained in step 2 is p_value (we will replace it with the actual value after calculation). If p_value is significantly higher than the threshold, such as 0.05, then the data supports the student government's claim. If p_value is smaller, the evidence suggests that the percentage of students in favor of increasing fees is less than 80%.

Draw conclusions

Compare the p_value with the significance level of 0.05. If p_value > 0.05, we can conclude that the data is consistent with the student government's claim and there is not enough evidence to reject the assertion. However, if p_value < 0.05, the data implies that the percentage of students who favor increasing fees might be less than 80%, suggesting further investigation.

Key concepts

Probability

Probability is a key concept in statistics that describes the likelihood of a particular event occurring. In our exercise, we're dealing with a situation where we're trying to understand how probable it is for 15 or fewer students out of a sample of 25 to favor the fee increase, given that 80% of all students supposedly support it.

In a binomial distribution, you have a fixed number of independent trials, and each trial has only two possible outcomes, such as "favor" or "not favor". The binomial probability formula helps us calculate the likelihood of a specific number of successful outcomes in these trials.

• The success probability (\(p\)) refers to the proportion of favorable outcomes, in this case, 0.80 (or 80%).
• The number of trials (\(n\)) is 25, which represents our sample size.
• Successful outcomes (\(k\)) are the number of students in favor, which can be 15 or fewer in this example.

This calculation can help indicate whether the actual sample result aligns with the broader population expectation.

Cumulative Distribution Function (CDF)

The Cumulative Distribution Function (CDF) is a pivotal concept in understanding the probabilities over a range of outcomes in a probability distribution. For our exercise, the CDF helps answer the question: What is the probability of obtaining 15 or fewer students favoring the issue?

The CDF is calculated by summing up the probabilities of all possible outcomes from 0 to 15 students favoring the increase.

• Mathematically, it's represented as:\[P(X \le 15) = \sum_{k=0}^{15} {25\choose k}(0.80)^k(1-0.80)^{25-k}\]
• This formula sums the probabilities of each scenario from 0 up to 15 students favoring the fee increase.
• A binomial calculator or statistical software commonly assists in these calculations to quickly find the cumulative probability.

The resulting cumulative probability gives us the likelihood of our observed or more extreme outcomes occurring if the student government's claims hold true.

Statistical Significance

Statistical significance helps us determine whether a result from our sample data is strong enough to suggest an effect or occurrence genuinely exists in the population, rather than being a fluke.

In this scenario, once we have the cumulative probability value (often termed as p_value), we compare it against a common statistical threshold, which is usually 0.05. This threshold helps decide whether the result is significant or not.

• If the p_value is greater than 0.05, it suggests that the sample data is consistent with the population assumption (80% in favor) and there isn't enough evidence to refute the student government's claim.
• If the p_value is less than 0.05, it indicates the sample data shows a statistically significant difference, meaning fewer than 80% of students might actually support the increase, highlighting the claim could be incorrect.
• Statistical significance does not prove the claim wrong or right but suggests whether further investigation is warranted.

This concept ensures that conclusions drawn from data are robust and less prone to errors from noise or randomness in the sample.