The Weather Underground reported that the mean amount of...

The Weather Underground reported that the mean amount of summer rainfall for the northeastern US is at least 11.52 inches. Ten cities in the northeast are randomly selected and the mean rainfall amount is calculated to be 7.42 inches with a standard deviation of 1.3 inches. At the $\alpha=0.05$ level, can it be concluded that the mean rainfall was below the reported average? What if $\alpha=0.01 ?$ Assume the amount of summer rainfall follows a normal distribution.

Key Concepts

Degrees of Freedom

Degrees of freedom refer to the number of independent values or quantities which can be assigned to a statistical distribution. In the context of a one-sample t-test, the degrees of freedom typically equal the sample size minus one and influence the shape of the t-distribution used in the test.

t-Distribution

The t-distribution is a probability distribution used in statistical inference, particularly in t-tests, when the sample size is small and the population standard deviation is unknown. It is similar to the normal distribution but with heavier tails, which accounts for the greater variability expected in small samples.

Significance Level

The significance level, represented by alpha (?), is the threshold for determining when to reject the null hypothesis. It is the probability of committing a Type I error – rejecting a true null hypothesis – and common values include 0.05 and 0.01. A lower significance level requires stronger evidence to reject the null hypothesis.

One-Sample t-Test

A one-sample t-test is a statistical procedure used to compare the mean of a single sample to a known or hypothesized population mean when the population standard deviation is unknown. It is appropriate when the underlying distribution of the data is normal and the sample size is small.

Null Hypothesis

The null hypothesis, often denoted as H0, is a statement that there is no effect or no difference, or that a parameter is equal to a specified value. It serves as the starting assumption for statistical testing and is only rejected if the data provide strong evidence against it.

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions or inferences about population parameters based on sample data. It involves formulating a null hypothesis, which represents a default assumption, and an alternative hypothesis, which represents a claim to be tested. Data are then used to determine whether there is sufficient evidence to reject the null hypothesis in favor of the alternative hypothesis.

Alternative Hypothesis

The alternative hypothesis, denoted as Ha or H1, is a statement that contradicts the null hypothesis. It represents the outcome that the test is designed to support, such as the parameter being greater than, less than, or not equal to a specified value in the null hypothesis.

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