ATM Withdrawals The manufacturer of a certain type of ATM...

ATM Withdrawals The manufacturer of a certain type of ATM machine reports that the mean ATM withdrawal is $\$ 60 .$ The manager of a convenience store with an ATM machine claims that mean withdrawal from his machine is less than this amount. He obtains a simple random sample of 35 withdrawals over the past year and finds the sample mean to be $\$ 52 .$ Assume that the population standard deviation is $\$ 13 .$ Test the manager's claim using either the classical approach or the $P$ -value approach at the $\alpha=0.05$ level of significance.

Key Concepts

P-value Approach

The p-value is the probability of observing a test statistic as extreme as, or more extreme than, the statistic computed from the sample data under the assumption that the null hypothesis is true. It quantifies the strength of evidence against the null hypothesis and is used to determine whether the sample provides sufficient evidence to reject it by comparing it to the significance level.

Test Statistic (Z-test)

A test statistic is a standardized value computed from sample data that measures how far the sample statistic is from the hypothesized population parameter in units of the standard error. When the population standard deviation is known, the Z-test is used for normally distributed data, allowing comparison of the sample mean against a known benchmark to see if the observed deviation is statistically significant.

Type I and Type II Errors

In hypothesis testing, a Type I error occurs when the null hypothesis is rejected when it is actually true, while a Type II error occurs when the null hypothesis is not rejected when it is actually false. Understanding these errors is crucial for assessing the risks involved with hypothesis testing and for making informed decisions regarding the interpretation of results.

Null and Alternative Hypotheses

In hypothesis testing, the null hypothesis (H0) is the default claim about a population parameter, often a status quo or a benchmark value, while the alternative hypothesis (H1) represents a claim to be tested against the null hypothesis. The alternative often reflects a directional claim, for example that the parameter is less than a specified value, which is central to determining the appropriate test and rejection criteria.

Significance Level

The significance level, denoted as ?, is the probability threshold below which the null hypothesis is rejected. It represents the risk of committing a Type I error, meaning that there is a predetermined acceptable level of false positives. This concept is critical because it guides the decision-making process in hypothesis testing.

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