A random sample of 202 business faculty members was asked if...

A random sample of 202 business faculty members was asked if there should be a required foreign language course for business majors. Of these sample members, 140 felt there was a need for a foreign language course. Test the hypothesis that at least $75 \%$ of all business faculty members hold this view. Use $\alpha=0.05$.

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical methodology used to decide whether there is enough evidence in a sample of data to infer that a certain condition holds for the entire population. It involves comparing observed data against a null hypothesis to determine if the observed effect is statistically significant.

Null and Alternative Hypotheses

In hypothesis testing, the null hypothesis typically represents a default or established claim about a population parameter, while the alternative hypothesis represents what the researcher aims to provide evidence for. For instance, asserting that at least 75% of the population holds a certain view would be framed as the null hypothesis, with the alternative hypothesis challenging this claim.

Significance Level

The significance level, denoted by ?, is the probability threshold used to determine whether the observed test results are sufficiently unlikely under the null hypothesis. A significance level of 0.05 indicates that there is a 5% risk of rejecting the null hypothesis when it is in fact true.

Sample Proportion

The sample proportion is the ratio of individuals in the sample exhibiting a particular characteristic relative to the total sample size. It serves as an estimate of the corresponding population proportion and is critical in making inferences about the population in hypothesis testing.

Standard Error and Normal Approximation

When dealing with proportions, the standard error quantifies the expected variability of the sample proportion around the true population proportion. Under conditions where the sample size is sufficiently large, the distribution of the sample proportion can be approximated by a normal distribution, facilitating the calculation of probabilities and critical values in hypothesis testing.

Test Statistic

The test statistic in the context of proportions is typically calculated using a z-test formula, which standardizes the difference between the observed sample proportion and the hypothesized population proportion. This statistic is then compared to a critical value from the normal distribution to decide whether to reject the null hypothesis.

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