A manufacturer of detergent claims that the contents of...

A manufacturer of detergent claims that the contents of boxes sold weigh on average at least 16 ounces. The distribution of weight is known to be normal, with a standard deviation of 0.4 ounce. A random sample of 16 boxes yielded a sample mean weight of 15.84 ounces. Test at the $10 \%$ significance level the null hypothesis that the population mean weight is at least 16 ounces.

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical procedure that allows one to make inferences about a population parameter based on sample data. It involves formulating a null hypothesis (which represents the status quo or a claim to be tested) and an alternative hypothesis (which represents a competing claim) in order to assess whether observed data provide sufficient evidence to reject the null hypothesis.

Null and Alternative Hypotheses

In hypothesis testing, the null hypothesis (H0) is a statement about a population parameter that is assumed true until evidence suggests otherwise, whereas the alternative hypothesis (H1) is a statement that contradicts the null hypothesis. In the context of a one-sided test, the alternative hypothesis specifies that the parameter is either less than or greater than a specific value, depending on the claim being tested.

Significance Level

The significance level, often denoted by ?, is the threshold for rejecting the null hypothesis. It represents the probability of making a Type I error, which occurs when the null hypothesis is incorrectly rejected. A significance level of 10% means that there is a 10% risk of rejecting the null hypothesis when it is actually true.

Test Statistic and p-value

The test statistic is a standardized value calculated from sample data that measures the degree to which the observed data deviates from the null hypothesis. The p-value is the probability of obtaining a test statistic at least as extreme as the one observed, under the assumption that the null hypothesis is true. These metrics are compared against the significance level to make a decision about the null hypothesis.

Normal Distribution Assumption and Z-test

When the underlying population distribution is normal and the standard deviation is known, the Z-test can be used for hypothesis testing. The Z-test relies on the properties of the normal distribution to calculate the probability of observing a sample mean given a particular population mean, making it a suitable choice for tests involving normally distributed data with known variance.

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