A random sample of size 8 drawn from a normal population...

A random sample of size 8 drawn from a normal population yielded the following results: $\bar{x}-280, s=46$. a. Test $H 0: \mu=250$ vs. Ha:\mu $>250 @ \alpha=0.05$. 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 8 drawn from a normal population yielded the following results: $\bar{x}-280, s=46$.
a. Test $H 0: \mu=250$ vs. Ha:\mu $>250 @ \alpha=0.05$.
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, denoted by alpha (?), is the threshold used in hypothesis testing to decide whether to reject the null hypothesis. It represents the maximum acceptable probability of committing a Type I error, meaning rejecting a true null hypothesis.

p-Value

The p-value represents the probability of obtaining a test statistic as extreme as, or more extreme than, the value observed, under the assumption that the null hypothesis is true. It provides a quantitative measure to assess the strength of evidence against the null hypothesis.

Student's t-Test

A Student's t-test is used when testing hypotheses about the mean of a normally distributed population when the sample size is small and population variance is unknown. The test statistic is calculated using the sample mean, the hypothesized population mean, the sample standard deviation, and the square root of the sample size.

Hypothesis Testing

This concept involves formulating a null hypothesis (H0) and an alternative hypothesis (Ha) about a population parameter, then using sample data to decide whether there is enough evidence to reject the null hypothesis in favor of the alternative. The process helps in determining if an observed effect is statistically significant or could have occurred by chance.

Degrees of Freedom

Degrees of freedom in hypothesis testing, particularly in a t-test, are determined by the sample size minus one (n-1). They influence the shape of the t-distribution, which is crucial for determining critical values and p-values.

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