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

step 1 Okay, so here our first P value is probability that Z is going to be less than negative 3 .33. F

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 to be $\bar{x}=70$. Consider the null hypothesis $H_0: \mu=80$ versus the alternative hypothesis $H_1: \mu<80$. Compute the $p$-value 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 to be $\bar{x}=70$. Consider the null hypothesis $H_0: \mu=80$ versus the alternative hypothesis $H_1: \mu<80$. Compute the $p$-value 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

p-Value

The p-Value is the probability of obtaining a test statistic at least as extreme as the one measured, given that the null hypothesis is true. In a one-tailed test, such as testing whether the mean is less than a given value, the p-value is computed as the probability that the test statistic falls in the tail of the standard normal distribution. A small p-value provides evidence against the null hypothesis.

Standard Error

The standard error measures the variability of the sample mean around the population mean and is calculated by taking the square root of the population variance divided by the sample size. It quantifies how much the sample mean is expected to vary from sample to sample, which is essential for computing the Z-score in a hypothesis test involving a known variance.

Null and Alternative Hypotheses

Hypothesis testing involves setting up a null hypothesis, which is a default statement about a population parameter, and an alternative hypothesis, which contradicts the null. In the context of testing a population mean, the null hypothesis typically states that the mean is equal to a specified value, while the alternative hypothesis states that it is either greater than, less than, or not equal to that value. This framework provides the structure for the test decision.

Z-Test

The Z-Test is a statistical method used to determine whether there is a significant difference between the sample mean and the hypothesized population mean when the population variance is known. The test converts the difference into a Z-score by dividing by the standard error, allowing the use of the standard normal distribution to assess the probability of the observed value under the null hypothesis.

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