A random sample of size 20 drawn from a normal population...

A random sample of size 20 drawn from a normal population yielded the following results: $\bar{x}-49.2,5=1.33$ a. Test $\boldsymbol{H}_{\mathrm{a}}: \mu-50$ vs. $\boldsymbol{H}_{\mathrm{a}}: \boldsymbol{\mu}+50 @ \mathrm{a}-\mathrm{O} .01 .$ 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 20 drawn from a normal population yielded the following
results: $\bar{x}-49.2,5=1.33$
a. Test $\boldsymbol{H}_{\mathrm{a}}: \mu-50$ vs. $\boldsymbol{H}_{\mathrm{a}}: \boldsymbol{\mu}+50 @ \mathrm{a}-\mathrm{O} .01 .$
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

t Distribution and Degrees of Freedom

The t-distribution is a family of continuous probability distributions that is used when dealing with small sample sizes and unknown population variance. The shape of the t-distribution is determined by its degrees of freedom, typically calculated as the sample size minus one (n-1), which reflects the amount of independent information used in estimating the variability of the sample mean.

Significance Level

The significance level, often denoted by alpha (?), is the threshold used to determine whether the observed p-value is sufficiently small to reject the null hypothesis. It represents the probability of committing a Type I error, which is the error of incorrectly rejecting a true null hypothesis. In hypothesis testing, if the p-value is less than or equal to the significance level, the null hypothesis is rejected.

Hypothesis Testing

Hypothesis testing is a statistical procedure used to evaluate assumptions about a population parameter based on sample data. It involves formulating two competing hypotheses, the null hypothesis (which typically represents a status quo or claim of no effect) and an alternative hypothesis (which represents a claim of effect or difference), and then using sample statistics to decide whether to reject the null hypothesis in favor of the alternative hypothesis.

One-sample t-test

The one-sample t-test is used to determine whether the mean of a sample significantly differs from a known or hypothesized value, particularly when the population standard deviation is unknown and the sample size is small. This test employs the t-distribution which accounts for additional uncertainty by estimating the variability from the sample itself.

p-Value

The p-value represents the probability of obtaining a test statistic as extreme as, or more extreme than, the value observed in the sample data, assuming that the null hypothesis is true. It serves as a measure of the strength of evidence against the null hypothesis, where a smaller p-value indicates stronger evidence to reject the null hypothesis.

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