Consider the following hypothesis test. H0: μ⩾10 Ha=μ<10 The sample size is 120 and

Consider the following hypothesis test. H0: μ⩾10 ...

Consider the following hypothesis test. $$ \begin{aligned} & H_0: \mu \geqslant 10 \\ & H_{\mathrm{a}}=\mu<10 \end{aligned} $$ The sample size is 120 and the population standard deviation is assumed known with $a=5$. Use $\alpha=.05$. a. If the population mean is 9 , what is the probability that the sample mean leads to the conclusion do not reject $\mathrm{H}_0$ ? b. What rype of error would be made if the actual population mean is 9 and we conclude that $H_0 ; \mu \geq 10$ is true? c. What is the probability of making a Type II error if the actual population mean is 8 ?

Consider the following hypothesis test.
$$
\begin{aligned}
& H_0: \mu \geqslant 10 \\
& H_{\mathrm{a}}=\mu<10
\end{aligned}
$$
The sample size is 120 and the population standard deviation is assumed known with $a=5$. Use $\alpha=.05$.
a. If the population mean is 9 , what is the probability that the sample mean leads to the conclusion do not reject $\mathrm{H}_0$ ?
b. What rype of error would be made if the actual population mean is 9 and we conclude that $H_0 ; \mu \geq 10$ is true?
c. What is the probability of making a Type II error if the actual population mean is 8 ?

Key Concepts

Type I and Type II Errors

In hypothesis testing, a Type I error occurs when the null hypothesis is wrongly rejected despite being true, while a Type II error happens when the test fails to reject a false null hypothesis. These error types are inherent in the testing process, and their probabilities are influenced by the chosen significance level (for Type I) and the power of the test (for Type II). Understanding these errors is critical for correctly interpreting test results and assessing the reliability of conclusions drawn from statistical analyses.

Z-test and Sampling Distribution of the Sample Mean

When the population standard deviation is known and the sample size is relatively large, the sampling distribution of the sample mean is approximately normal (by the Central Limit Theorem). In such cases, a Z-test is employed to assess the evidence against the null hypothesis. The Z-test statistic is calculated by standardizing the difference between the observed sample mean and the hypothesized population mean, allowing the use of the standard normal distribution to evaluate the probability of observing such a statistic under the null hypothesis.

Power of a Test

The power of a test is the probability that the test correctly rejects a false null hypothesis; in other words, it measures the test's ability to detect an effect when one truly exists. A higher power indicates a lower probability of making a Type II error. Factors influencing the power include the sample size, the effect size (the true difference), and the chosen significance level, making it an essential concept in designing and interpreting hypothesis tests.

Hypothesis Testing

Hypothesis testing is a statistical method used to decide whether there is enough evidence in a sample of data to infer that a certain condition holds for the entire population. This involves formulating two competing hypotheses, the null hypothesis (which represents a default position or status quo) and the alternative hypothesis (which represents a differing claim). Decisions are made based on the data collected and pre-determined significance levels, ensuring that conclusions are supported by probability theory.

Significance Level and Critical Region

The significance level, denoted by ?, is the threshold probability for rejecting the null hypothesis when it is actually true. It defines the critical region, which is the set of outcomes that are sufficiently unlikely under the assumption of the null hypothesis. By comparing the test statistic to a critical value derived from a theoretical distribution, researchers can decide whether to reject or fail to reject the null hypothesis based on the observed data.

Transcript

00:01 Hello, so here we have our hypotheses as the null hypothesis is going to be mu is greater than equal to 10, and then our alternative hypothesis is going to be mu is less than 10.

00:10 So here we have our sample n is equal to 120, and our standard deviation, sigma, is 5, with alpha equal to 0 .05.

00:20 So our test statistic is then equal to x bar minus mu not.

00:24 So that's going to be x bar minus 10 divided by sigma, so 5 divided by the square root of 120.

00:35 And based on the critical value approach, we get that z sub 0 .05 is equal to 1 .645.

00:44 So the rejection rule for the lower tail is that we reject h not if z is less than or equal to negative 1 .640.

00:55 5.

00:56 So here we get that z is going to be equal to, again, we found right here.

01:02 Solving for x bar, we get that x bar is going to be equal to, well, x bar is less than 10 minus 1 .645 times 5 over square root of 120, which is going to be equal to 9 .25.

01:20 So therefore we reject h0 whenever x bar is less than or equal to 9 .25.

01:26 This means that we made the conclusion to not reject h0 whenever x bar is greater than 9 .25.

01:35 So we take the population mean here, mu to be 9.

01:39 And if mu 9 is really true, then the probability of accepting h0 when mu is greater than equal to 10 is equal to the probability.

01:50 That the sample mean x bar is greater than 9 .25 when mu is equal to 9...

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