The manager of a restaurant in a large city claims that...

The manager of a restaurant in a large city claims that waiters working in all restaurants in his city eart an average of $\$ 150$ or more in tips per week. A random sample of 25 waiters selected from restaurants of this city yielded a mean of $\$ 139$ in tips per week with a standard deviation of $\$ 28$. Assume that the weekly tips for all waiters in this city have a normal distribution a. Using a $1 \%$ significance level, can you conclude that the manager's claim is true? Use both approaches b. What is the Type I error in this exercise? Explain. What is the probability of making such an error'

The manager of a restaurant in a large city claims that waiters working in all restaurants in his city eart
an average of $\$ 150$ or more in tips per week. A random sample of 25 waiters selected from restaurants of this city yielded a mean of $\$ 139$ in tips per week with a standard deviation of $\$ 28$. Assume that the weekly tips for all waiters in this city have a normal distribution
a. Using a $1 \%$ significance level, can you conclude that the manager's claim is true? Use both approaches
b. What is the Type I error in this exercise? Explain. What is the probability of making such an error'

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to decide whether there is enough evidence in a sample of data to infer that a certain condition holds true for the entire population. In this context, it involves formulating a null hypothesis (typically that the average tips are at least $150) and an alternative hypothesis (that they are less than $150), then using sample data to decide whether to reject the null hypothesis.

Significance Level

The significance level is the probability threshold set for rejecting the null hypothesis when it is actually true, representing the risk of a Type I error. In this problem, a 1% significance level is used, meaning that there is a 1% risk of incorrectly concluding that the restaurant manager’s claim is false when it is in fact true.

Test Statistic and t-Distribution

When the population standard deviation is unknown and the sample size is small, the t-distribution is used to calculate the test statistic, which measures how far the sample mean is from the hypothesized population mean relative to the estimated standard error. This test statistic is compared against critical values from the t-distribution corresponding to the chosen significance level.

Critical Value and p-Value Approaches

There are two common approaches in hypothesis testing: the critical value approach, which involves comparing the calculated test statistic to a threshold value that marks the boundary of the rejection region, and the p-value approach, which involves calculating the probability of obtaining a result as extreme or more extreme than the observed one under the null hypothesis. Both methods help in deciding whether to reject the null hypothesis.

Type I Error

A Type I error occurs when the null hypothesis is incorrectly rejected while it is actually true. In the context of this exercise, the Type I error would be rejecting the claim that waiters earn an average of $150 or more in tips when in reality they do earn that amount. The probability of committing this error is equal to the significance level, which in this case is 1%.

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