Unfortunately, arsenic occurs naturally in some ground water...

Unfortunately, arsenic occurs naturally in some ground water (Reference: Union Carbide Technical Report $K / U R-1$ ). A mean arsenic level of $\mu=8.0$ parts per billion (ppb) is considered safe for agricultural use. A well in Texas is used to water cotton crops. This well is tested on a regular basis for arsenic. A random sample of 37 tests gave a sample mean of $\bar{x}=7.2 \mathrm{ppb}$ arsenic, with $s=1.9 \mathrm{ppb} .$ Does this information indicate that the mean level of arsenic in this well is less than 8 ppb? Use $\alpha=0.01.$

Key Concepts

One-Tailed Test

A one-tailed test is used when the alternative hypothesis specifies a direction of the effect (for example, that the mean is less than or greater than the hypothesized value). It examines the probability of deviating from the hypothesized parameter in one direction only, as opposed to two-tailed tests which consider both directions.

Significance Level

The significance level, denoted as alpha (?), is the threshold for deciding when to reject the null hypothesis. It represents the probability of making a Type I error, which is rejecting the null hypothesis when it is actually true. Common values for ? are 0.05, 0.01, etc., indicating the level of risk the researcher is willing to accept.

t-Distribution and Degrees of Freedom

The t-distribution is used in hypothesis testing for means when the sample size is small and the population variance is unknown. The shape of the t-distribution is influenced by the degrees of freedom, which is typically the sample size minus one in a one-sample t-test. As the degrees of freedom increase, the t-distribution approaches the normal distribution.

Test Statistic and p-Value

The test statistic in a t-test measures how far the sample statistic is from the hypothesized population mean in terms of standard errors. The p-value is then computed based on this statistic; it represents the probability of obtaining a result as extreme as, or more extreme than, the one observed, under the assumption that the null hypothesis is true. A small p-value suggests that the observed data is unlikely under the null hypothesis, leading to its rejection.

Null and Alternative Hypotheses

The null hypothesis typically states that there is no difference or no effect, while the alternative hypothesis reflects the claim we want to test against the status quo. In tests concerning population means, the null hypothesis might state that the mean equals a specific value, while the alternative could state that the mean is less than, greater than, or not equal to that value.

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 for the entire population. It involves formulating a null hypothesis (a statement of no effect or status quo) and an alternative hypothesis (a statement of effect or difference), and then using sample data to decide whether to reject the null hypothesis.

One-Sample t-Test

A one-sample t-test is a statistical procedure used to determine whether the mean of a single sample differs significantly from a known or hypothesized population mean. This test is especially useful when the sample size is relatively small and the population standard deviation is unknown, meaning the sample standard deviation is used instead.

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