A statistics instructor believes that fewer than 20 % of...

A statistics instructor believes that fewer than $20 \%$ of Evergreen Valley College (EVC) students attended the opening night midnight showing of the latest Harry Potter movie. She surveys 84 of her students and finds that 11 attended the midnight showing. An appropriate alternative hypothesis is: a. $p=0.20$ b. $p>0.20$ c. $p<0.20$ d. $p \leq 0.20$

A statistics instructor believes that fewer than $20 \%$ of Evergreen Valley College (EVC) students attended the opening night midnight showing of the latest Harry Potter movie. She surveys 84 of her students and finds that 11 attended the midnight showing. An appropriate alternative hypothesis is:
a. $p=0.20$
b. $p>0.20$
c. $p<0.20$
d. $p \leq 0.20$

Key Concepts

Hypothesis Testing

Hypothesis testing is a formal procedure used to assess a claim about a population parameter using sample data. The process involves setting up a null hypothesis, which represents a default or status quo claim, and an alternative hypothesis, which represents the research or claim of interest. Decisions are made based on how likely the observed data is under these hypotheses.

Null Hypothesis

The null hypothesis is a statement about a population parameter that asserts no effect or no difference from a specific value. In tests dealing with proportions, it typically claims that the parameter is equal to a specified value. Its role is to serve as the default assumption that any observed deviations in the sample are due to random variation.

Alternative Hypothesis

The alternative hypothesis is the statement that contradicts the null hypothesis by incorporating the researcher's expectations or claims. It specifies a condition, such as the proportion being either greater than or less than a specific value, which directs the test towards a particular outcome. The alternative hypothesis helps to determine if the observed effect is statistically significant.

One-Tailed Test

A one-tailed test is used when the alternative hypothesis predicts the direction of the effect—either that a parameter is less than or greater than the hypothesized value. This approach focuses on a single area of the distribution, thereby increasing the test’s power to detect an effect in a specific direction. In the context of the given question, the claim that fewer than 20% of students attended the event leads to using a one-tailed test where the alternative hypothesis is that the true proportion is less than 0.20.

Population Proportion

Population proportion is the fraction or percentage of the total population that possesses a particular characteristic. In hypothesis testing, the population proportion is the parameter being tested to determine if it matches a stated value, based on data sampled from the population. This concept is central to problems involving proportions, like the one described in the question.

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