A random sample of n=25 is obtained from a population with variance σ^2, and the sample mean is

step 1 Hello. So here we have that our null hypothesis is going to be that mu is equal to 100, and then

A random sample of n=25 is obtained from a population with...

A random sample of $n=25$ is obtained from a population with variance $\sigma^2$, and the sample mean is computed. Test the null hypothesis $H_0: \mu=100$ versus the alternative hypothesis $H_1: \mu>100$ with $\alpha=0.05$. Compute the critical value $\bar{x}_c$ and state your decision rule for the following options. a. The population variance is $\sigma^2=225$. b. The population variance is $\sigma^2=900$. c. The population variance is $\sigma^2=400$. d. The population variance is $\sigma^2=600$.

A random sample of $n=25$ is obtained from a population with variance $\sigma^2$, and the sample mean is computed. Test the null hypothesis $H_0: \mu=100$ versus the alternative hypothesis $H_1: \mu>100$ with $\alpha=0.05$.
Compute the critical value $\bar{x}_c$ and state your decision rule for the following options.
a. The population variance is $\sigma^2=225$.
b. The population variance is $\sigma^2=900$.
c. The population variance is $\sigma^2=400$.
d. The population variance is $\sigma^2=600$.

Key Concepts

Critical Value

The critical value is a cutoff point from the standard normal distribution that corresponds to a chosen significance level (alpha). In a one-sided test, it defines the threshold beyond which the observed test statistic leads to the rejection of the null hypothesis, ensuring control over the Type I error rate.

Decision Rule

The decision rule in hypothesis testing is the predetermined guideline for rejecting or failing to reject the null hypothesis based on the comparison between the test statistic and the critical value. For a one-sided test, if the calculated test statistic exceeds the critical value, then the null hypothesis is rejected in favor of the alternative hypothesis.

Standard Error

The standard error measures the dispersion of the sampling distribution of the sample mean. It is computed by taking the square root of the population variance divided by the sample size. This value is critical in standardizing the difference between the sample mean and the hypothesis mean in the z-test.

Z-test

A z-test is used when the population variance is known or the sample size is sufficiently large for the Central Limit Theorem to apply. The test statistic is calculated by standardizing the difference between the sample mean and the hypothesized mean by dividing by the standard error, which is derived from the known variance and the sample size.

Known Population Variance

Knowing the population variance allows for the use of tests that rely on precise characterization of the sampling distribution. When the variance is known, the exact form of the sampling distribution—typically normal—can be employed, which simplifies the calculation of test statistics and critical values.

Null and Alternative Hypotheses

The null hypothesis (H0) represents a default or initial claim about a population parameter, such as a specific mean value, while the alternative hypothesis (H1) is what one seeks evidence to support, indicating a deviation from the null. Setting up these hypotheses correctly is crucial for determining the direction and nature of any statistical test.

Hypothesis Testing

Hypothesis testing is a statistical process used to evaluate a claim about a population parameter by comparing sample evidence with what is expected under a specific assumption, known as the null hypothesis. It provides a systematic way to decide whether the observed data are consistent with a given hypothesis about the population.

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