For a sample of 12 items from a normally distributed...

For a sample of 12 items from a normally distributed population for which the standard deviation is $\sigma=17.0$, the sample mean is 230.8 . At the 0.05 level of significance, test $H_0: \mu \leq 220$ versus $H_1: \mu>220$. Determine and interpret the $p$-value for the test.

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to decide between two competing claims (hypotheses) about a population parameter based on sample data. It involves setting up a null hypothesis, which represents a baseline or a default condition, and an alternative hypothesis, which represents a claim for change or difference. Decisions are made by comparing the evidence provided by the sample against the expectations under the null hypothesis.

Significance Level

The significance level, denoted by alpha (?), defines the probability of committing a Type I error – that is, rejecting the null hypothesis when it is actually true. It sets a threshold against which the p-value is compared. Commonly used values are 0.05 or 0.01, indicating a 5% or 1% risk of Type I error, respectively.

Test Statistic

A test statistic is a standardized value calculated from sample data during a hypothesis test. In cases where the population standard deviation is known and the data are normally distributed, the z-test statistic is used, which is computed by subtracting the null hypothesis mean from the sample mean and dividing by the standard error of the sample mean. This statistic indicates how many standard deviations the sample mean is away from the null hypothesis mean.

p-value

The p-value is the probability, assuming that the null hypothesis is true, of obtaining a test statistic at least as extreme as the one that was actually observed. It provides a measure of the evidence against the null hypothesis; a smaller p-value indicates stronger evidence in favor of the alternative hypothesis. When the p-value is less than or equal to the chosen significance level, the null hypothesis is rejected.

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