A pharmaceutical manufacturer is concerned that the impurity...

A pharmaceutical manufacturer is concerned that the impurity concentration in pills does not exceed $3 \%$. It is known that from a particular production run, impurity concentrations follow a normal distribution with standard deviation $0.4 \%$. A random sample of 64 pills from a production run was checked, and the sample mean impurity concentration was found to be $3.07 \%$. a. Test, at the $5 \%$ level, the null hypothesis that the population mean impurity concentration is $3 \%$ against the alternative that it is more than $3 \%$. b. Find the probability of a $5 \%$-level test rejecting the null hypothesis when the true mean impurity concentration is $3.10 \%$.

A pharmaceutical manufacturer is concerned that the impurity concentration in pills does not exceed $3 \%$. It is known that from a particular production run, impurity concentrations follow a normal distribution with standard deviation $0.4 \%$. A random sample of 64 pills from a production run was checked, and the sample mean impurity concentration was found to be $3.07 \%$.
a. Test, at the $5 \%$ level, the null hypothesis that the population mean impurity concentration is $3 \%$ against the alternative that it is more than $3 \%$.
b. Find the probability of a $5 \%$-level test rejecting the null hypothesis when the true mean impurity concentration is $3.10 \%$.

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to decide whether the evidence provided by a sample is strong enough to reject a specified claim about a population parameter, known as the null hypothesis. This process involves formulating both a null hypothesis and an alternative hypothesis, computing a test statistic, and comparing it to a critical value or calculating a p-value to make the decision.

Null and Alternative Hypotheses

The null hypothesis (H?) is a statement asserting that there is no effect or no difference in the population parameter, while the alternative hypothesis (H?) represents what the test aims to provide evidence for, typically indicating a change or difference from the null condition. In testing, H? is assumed to be true until the evidence suggests otherwise.

Significance Level

The significance level, often denoted by alpha (?), is the threshold used to determine whether a test result is statistically significant. It is the probability of rejecting the null hypothesis when it is actually true (Type I error). A common significance level is 5%, meaning there is a 5% risk of incorrectly rejecting a true null hypothesis.

One-Sample Z-Test

A one-sample Z-test is a hypothesis test used when comparing a sample mean to a known population mean, provided that the population standard deviation is known. It involves standardizing the difference between the sample mean and the population mean using the standard error, then comparing the resulting Z-score to critical values from the standard normal distribution.

Sampling Distribution of the Mean

The sampling distribution of the mean is the probability distribution of sample means drawn from a population. According to the Central Limit Theorem, if the sample size is sufficiently large, this distribution will be approximately normal, regardless of the shape of the population distribution, which justifies the use of Z-tests in many practical scenarios.

Power of a Test

The power of a test is the probability that the test correctly rejects a false null hypothesis, representing the test's ability to detect an effect when it truly exists. High power reduces the risk of a Type II error (failing to reject a false null hypothesis) and is influenced by factors such as the sample size, the true effect size, the standard deviation, and the chosen significance level.

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