A student is tested to determine whether she can tell butter...

A student is tested to determine whether she can tell butter from margarine. She is blindfolded and given small bites of toast that has been spread with either butter or margarine that have been randomly chosen. The experiment is designed so that she will have exactly 15 bites with butter and 15 bites with margarine. She gets 20 right out of 30 trials. Can she tell butter from margarine at a $0.05$ level of significance? Explain.

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to decide whether observed data sufficiently contradict a stated hypothesis, typically the null hypothesis. It involves comparing the observed data against a model assumed under the null hypothesis, using a test statistic to determine whether any deviation is likely due to chance.

Null and Alternative Hypotheses

In hypothesis testing, the null hypothesis typically represents a default position of no effect or no difference, while the alternative hypothesis represents the change or effect that one is testing for. Establishing these hypotheses allows for a systematic evaluation of whether the evidence is strong enough to reject the null hypothesis in favor of the alternative.

Binomial Distribution

The binomial distribution is a probability distribution that summarizes the likelihood of a given number of successes in a fixed number of independent trials, where each trial has a constant probability of success. It is particularly useful in experiments where there are only two possible outcomes (such as success or failure) and can be used to calculate the exact probability of observing a specific number of successes.

Significance Level and P-Value

The significance level, often denoted by ?, defines the threshold for rejecting the null hypothesis; it represents the probability of making a Type I error, which is rejecting the null hypothesis when it is actually true. The p-value, on the other hand, is the probability, under the null hypothesis, of obtaining a result as extreme as or more extreme than the observed result. Comparing the p-value to the significance level helps determine whether the observed effect is statistically significant.

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