Northeastern Franchisers, Ltd., has a number of clients that...

Northeastern Franchisers, Ltd., has a number of clients that use their process for producing exotic Norwegian dinners for customers throughout New England. The operating cost for the franchised process has a fixed cost of $$\$ 1,000$$ per week plus $$\$ 5$$ for $\mathrm{ev}$ ery unit produced. Recently, a number of restaurant owners using the process have complained that the cost model is no longer valid and, in fact, the weekly costs are higher. Your job is to determine if there is strong evidence to support the owners' claim. To do so, you obtain a random sample of $n=25$ restaurants and determine their costs. You also find that the number of units produced in each restaurant is normally distributed with a mean of $\mu=400$ and a variance of $\sigma^2=625$. The random sample mean $(n=25)$ for weekly costs was $\$ 3,050$. Prepare and implement an analysis to determine if there is strong evidence to conclude that costs are greater than those predicted by the cost model.

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to decide whether there is enough evidence to reject a null hypothesis about a population parameter. In the context of this problem, it involves assessing observed data against a cost model prediction to determine if there is a statistically significant difference in costs.

Null and Alternative Hypotheses

The null hypothesis (H?) represents a default position that there is no effect or no difference; in this case, that the actual costs are equal to those predicted by the cost model. The alternative hypothesis (H?) represents the claim that the costs are actually higher than predicted. This formulation sets up a one-sided test to see if the costs exceed the model's estimate.

Test Statistic

The test statistic is a standardized value that is calculated from sample data during a hypothesis test. When population variance is known or approximated and the sample size is sufficiently large (or the population is normally distributed), a Z-test statistic is used. It allows comparison of the observed sample mean to the hypothesized mean by scaling the difference with the standard error.

Sampling Distribution

The sampling distribution of the sample mean describes the probability distribution of the mean for samples of a given size taken from the population. In this problem, the production cost per restaurant follows a normal distribution, so the sampling distribution of the sample mean will also be normal with a mean equal to the true mean and a variance equal to the original variance divided by the sample size.

P-value

The p-value is the probability of obtaining a test statistic at least as extreme as the one observed, assuming that the null hypothesis is true. It provides a measure of the strength of the evidence against the null hypothesis. A low p-value indicates that such an extreme observed outcome is unlikely under the null hypothesis, leading to its potential rejection.

Significance Level

The significance level, often denoted by alpha (?), is the threshold for rejecting the null hypothesis. It represents the probability of making a Type I error, where a true null hypothesis is incorrectly rejected. In hypothesis testing, comparing the p-value to the significance level helps in making a decision regarding the validity of the null hypothesis.

Type I and Type II Errors

A Type I error occurs when the null hypothesis is wrongly rejected when it is in fact true, while a Type II error occurs when the null hypothesis is not rejected when the alternative hypothesis is true. Understanding these errors is crucial for setting an appropriate significance level and for evaluating the reliability of a hypothesis test's conclusion.

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