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 of them attended the midnight showing. At a $1 \%$ level of significance, an appropriate conclusion is: a. There is insufficient evidence to conclude that the percent of EVC students who attended the midnight showing of Harry Potter is less than $20 \%$. b. There is sufficient evidence to conclude that the percent of EVC students who attended the midnight showing of Harry Potter is more than $20 \%$. c. There is sufficient evidence to conclude that the percent of EVC students who attended the midnight showing of Harry Potter is less than $20 \%$. d. There is insufficient evidence to conclude that the percent of EVC students who attended the midnight showing of Harry Potter is at least $20 \%$

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 of them attended the midnight showing. At a $1 \%$ level of significance, an appropriate conclusion is:
a. There is insufficient evidence to conclude that the percent of EVC students who attended the midnight showing
of Harry Potter is less than $20 \%$.
b. There is sufficient evidence to conclude that the percent of EVC students who attended the midnight showing of Harry Potter is more than $20 \%$.
c. There is sufficient evidence to conclude that the percent of EVC students who attended the midnight showing of Harry Potter is less than $20 \%$.
d. There is insufficient evidence to conclude that the percent of EVC students who attended the midnight showing
of Harry Potter is at least $20 \%$

Key Concepts

Significance Level

The significance level, denoted by alpha (?), is the threshold for making a decision in hypothesis testing. It represents the probability of rejecting the null hypothesis when it is actually true (Type I error). A lower significance level means that stronger evidence is needed before rejecting the null hypothesis. This level is predetermined by the researcher and is compared to the p-value derived from the test.

One-Tailed (Directional) Test

A one-tailed test is used when the alternative hypothesis specifies a direction of the effect—for instance, that a population proportion is either higher or lower than the value stated in the null hypothesis. This type of test assesses the probability of the observed statistic occurring under the null hypothesis in only one direction, making it more powerful when the direction of the effect is specified in advance.

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions or inferences about a population based on sample data. It involves formulating a null hypothesis, which represents a default position or a statement of no effect, and an alternative hypothesis, which represents the claim to be tested. The process then uses sample data to determine whether there is enough evidence to reject the null hypothesis in favor of the alternative.

Null and Alternative Hypotheses

The null hypothesis (H?) is a statement about a population parameter that indicates no change or no effect, while the alternative hypothesis (H?) reflects the claim or suspicion that the parameter differs in a specific direction. In tests concerning population proportions, the null hypothesis typically states that the proportion equals a given value, whereas the alternative hypothesis specifies that the proportion is either less than, greater than, or not equal to that value.

Test Statistic and Critical Value

The test statistic is a standardized value computed from sample data that measures the degree to which the sample data diverges from the null hypothesis. In hypothesis tests for a proportion, the test statistic often follows a normal distribution when sample sizes are large enough. The critical value is the cutoff point(s) determined by the significance level; if the test statistic exceeds this cutoff (or falls in the critical region), the null hypothesis is rejected.

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