A random sample of size 12 drawn from a normal population...

A random sample of size 12 drawn from a normal population yielded the following results: $x-=86.2, s=0.63$. a. Test $H 0: \mu=85.5$ vs. Ha:\mu $\neq 85.5 @ \alpha=0.01$. b. Estimate the observed significance of the test in part (a) and state a decision based on the $p$ -value approach to hypothesis testing.

A random sample of size 12 drawn from a normal population yielded the following results: $x-=86.2, s=0.63$.
a. Test $H 0: \mu=85.5$ vs. Ha:\mu $\neq 85.5 @ \alpha=0.01$.
b. Estimate the observed significance of the test in part (a) and state a decision based on the $p$ -value approach to hypothesis testing.

Key Concepts

Significance Level

The significance level, commonly denoted as alpha (?), is the threshold for rejecting the null hypothesis. It represents the probability of committing a Type I error, which is the mistake of rejecting a true null hypothesis. Common choices for alpha are 0.05 or 0.01, and the selected level directly impacts the critical region of the test.

p-Value Approach

The p-value is the probability, assuming that the null hypothesis is true, of obtaining a test statistic at least as extreme as the one observed. In hypothesis testing, the p-value is compared to the significance level; if the p-value is less than alpha, the null hypothesis is rejected. This approach provides an exact measure of evidence against the null hypothesis.

Degrees of Freedom

Degrees of freedom in the context of a one-sample t-test refer to the number of independent values that are free to vary when estimating a parameter. For a sample of size n, the degrees of freedom are n - 1. They play a crucial role in determining the shape of the t-distribution, which is used to interpret the test statistic.

Decision Rule

The decision rule in hypothesis testing involves comparing the computed test statistic or p-value to the critical value or significance level, respectively. If the test statistic falls in the critical region or if the p-value is less than the predetermined significance level, the null hypothesis is rejected, indicating that the observed data is statistically significant.

One-Sample t-Test

The one-sample t-test is used to determine whether the mean of a single sample differs significantly from a known or hypothesized population mean when the population standard deviation is unknown and the sample size is small. It relies on the t-distribution, which adjusts for the additional uncertainty introduced by estimating the standard deviation from the sample.

Null and Alternative Hypotheses

The null hypothesis (H?) is a statement about a population parameter that is assumed to be true unless evidence indicates otherwise, while the alternative hypothesis (H?) is a statement that contradicts the null and represents what the data is tested against. These hypotheses form the basis of statistical tests in order to determine if any observed effect in the data is statistically significant.

Hypothesis Testing

Hypothesis testing is a statistical procedure used to make inferences or draw conclusions about a population based on sample data. It involves formulating a null hypothesis (a default position) and an alternative hypothesis, then using a test statistic calculated from the data to decide whether the sample provides enough evidence to reject the null hypothesis in favor of the alternative.

Test Statistic Calculation

The test statistic in a one-sample t-test is calculated by taking the difference between the sample mean and the hypothesized population mean, and then dividing that difference by the standard error of the mean. This calculation standardizes the difference in units of standard error, allowing comparison with the t-distribution to assess the significance of the result.

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