A random sample of size 16 drawn from a normal population...

A random sample of size 16 drawn from a normal population yielded the following results: $\bar{x}--0.06, s=1.07$. a. Test $\boldsymbol{H}_{\mathrm{a}}: \mu-\mathrm{O}$ vs. $\boldsymbol{H}_{\mathrm{a}}: \mu<\mathrm{O} @ \mathrm{a}-0.001 .$ 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 16 drawn from a normal population yielded the following results: $\bar{x}--0.06, s=1.07$.
a. Test $\boldsymbol{H}_{\mathrm{a}}: \mu-\mathrm{O}$ vs. $\boldsymbol{H}_{\mathrm{a}}: \mu<\mathrm{O} @ \mathrm{a}-0.001 .$
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

p-value Approach

The p-value is the probability of observing a test statistic as extreme as, or more extreme than, the observed value under the assumption that the null hypothesis is true. In the p-value approach to hypothesis testing, if the p-value is less than or equal to the chosen significance level, the null hypothesis is rejected; otherwise, it is not rejected.

Significance Level (alpha)

The significance level, denoted as alpha, represents the threshold for deciding whether the observed data is too unlikely under the null hypothesis. It is the probability of committing a Type I error—that is, rejecting the null hypothesis when it is true. Setting a very low alpha, such as 0.001, indicates a stringent criterion for rejecting the null hypothesis.

One-tailed Test

A one-tailed test is used when the alternative hypothesis specifies a direction of the effect (e.g., the mean is less than a certain value). This approach tests for the probability of outcomes in one tail of the distribution only, which is appropriate when deviations in a specific direction are of interest.

One-sample t-test

The one-sample t-test is used to compare the sample mean to a known or hypothesized value when the population standard deviation is unknown and the sample is drawn from a normally distributed population. It is appropriate when working with small sample sizes, and it accounts for the added uncertainty by using the t-distribution instead of the normal distribution.

Null and Alternative Hypotheses

In hypothesis testing, the null hypothesis represents the status quo or a statement of no effect, and the alternative hypothesis represents the claim to be tested. In tests involving means, the null hypothesis might state that the population mean equals a specific value, while the alternative hypothesis suggests a deviation (either one-tailed or two-tailed) from that value. The framing of these hypotheses guides the choice of test and interpretation of results.

t-distribution and Degrees of Freedom

The t-distribution is similar to the normal distribution but has heavier tails, which makes it more appropriate for small sample sizes. The degrees of freedom, typically calculated as the sample size minus one, determine the specific shape of the t-distribution. The correct degrees of freedom must be used when computing the t-statistic and obtaining critical values or p-values.

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