Consider the following hypothesis test: Ho: μ≤25 Ha: μ>25 A sample of 40 provided a

step 1 So from the given data here, so for part A, we compute the test statistic. So that's going to be

Consider the following hypothesis test: Ho: μ≤25 ...

Consider the following hypothesis test: $$ \begin{aligned} & H_{\mathrm{o}}: \mu \leq 25 \\ & H_{\mathrm{a}}: \mu>25 \end{aligned} $$ A sample of 40 provided a sample mesan of 26.4. The population standard deviation is 6 . a. Compute the value of the test statistic. b. What is the $p$-value? c. At $\alpha=.01$, what is your conclusion? d. What is the rejection rule using the critical value? What is your conclusion?

Consider the following hypothesis test:
$$
\begin{aligned}
& H_{\mathrm{o}}: \mu \leq 25 \\
& H_{\mathrm{a}}: \mu>25
\end{aligned}
$$
A sample of 40 provided a sample mesan of 26.4. The population standard deviation is 6 .
a. Compute the value of the test statistic.
b. What is the $p$-value?
c. At $\alpha=.01$, what is your conclusion?
d. What is the rejection rule using the critical value? What is your conclusion?

Key Concepts

Significance Level and Decision Making

The significance level (?) is a threshold set by the researcher that defines the probability of rejecting the null hypothesis when it is in fact true (Type I error). Common significance levels are 0.05, 0.01, etc. When the p-value is less than or equal to the significance level, it leads to rejection of the null hypothesis, implying that the observed effect is statistically significant.

p-Value

The p-value is the probability, under the null hypothesis, of obtaining a test statistic at least as extreme as the one that was actually observed. It quantifies the evidence against the null hypothesis; a smaller p-value indicates stronger evidence to reject the null hypothesis. In one-tailed tests, it reflects the probability of the sample result occurring in only one tail of the distribution.

Test Statistic

The test statistic quantifies how far the sample statistic is from the hypothesized parameter, relative to the standard error of the statistic. In the context of a one-sample z-test, it is calculated using the formula (x? - ??) / (? / ?n), where x? is the sample mean, ?? is the hypothesized population mean, ? is the population standard deviation, and n is the sample size. This statistic is then compared to the critical values from the standard normal distribution.

Critical Value and Rejection Rule

The critical value is the value from the test statistic's distribution that corresponds to the chosen significance level. For a one-tailed test, the rejection rule is established such that if the test statistic exceeds (or is less than, depending on the alternative hypothesis direction) this critical value, then the null hypothesis is rejected. This method provides an alternative means to decide between accepting or rejecting the null hypothesis by comparing the computed test statistic directly with the critical threshold.

Null and Alternative Hypotheses

The null and alternative hypotheses form the basis of any hypothesis test. The null hypothesis (H?) usually represents a default claim or a status quo, while the alternative hypothesis (H?) represents the claim that is investigated and is typically what a researcher wants to support. These hypotheses allow for a structured decision-making process based on sample data.

Hypothesis Testing

Hypothesis testing is a statistical method used to make inferences or decisions about a population parameter based on sample data. It involves setting up a null hypothesis (a statement of no effect or no difference) and an alternative hypothesis (a statement that contradicts the null), then using a test statistic to determine whether there is enough evidence to reject the null hypothesis in favor of the alternative.

One-Sample z-Test

A one-sample z-test is used to test hypotheses about a population mean when the population standard deviation is known and the sample size is sufficiently large, or the population is normally distributed. This test compares the sample mean to a hypothesized population mean using the standard normal distribution to determine the likelihood of observing a result as extreme as, or more extreme than, the sample result.

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