Answer the questions: (a) Is this a one- or two-tailed test?...

Answer the questions: (a) Is this a one- or two-tailed test? (b) What is the decision rule? (c) What is the value of the test statistic? (d) What is your decision regarding $H_{0} ?$ (e) What is the $p$ -value? Interpret it. A sample of 64 observations is selected from a normal population. The sample mean is $215,$ and the population standard deviation is $15 .$ Conduct the following test of hypothesis using the .025 significance level. $$ \begin{array}{l} H_{0}: \mu \geq 220 \\ H_{1}: \mu<220 \end{array} $$

Answer the questions: (a) Is this a one- or two-tailed test? (b) What is the decision rule? (c) What is the value of the test statistic? (d) What is your decision regarding $H_{0} ?$ (e) What is the $p$ -value? Interpret it.
A sample of 64 observations is selected from a normal population. The sample mean is $215,$ and the population standard deviation is $15 .$ Conduct the following test of hypothesis using the .025 significance level.
$$
\begin{array}{l}
H_{0}: \mu \geq 220 \\
H_{1}: \mu<220
\end{array}
$$

Key Concepts

Test Statistic

The test statistic is a standardized value calculated from the sample data that measures the extent of deviation from the null hypothesis. When the population standard deviation is known and the population is normally distributed, the z-test is commonly used. It incorporates the sample mean, the hypothesized population mean, and the standard error to determine how many standard deviations the sample mean is from the value stated under H0.

Null and Alternative Hypotheses

In hypothesis testing, the null hypothesis (H0) represents the status quo or a statement of no effect, which is assumed to be true until evidence indicates otherwise. The alternative hypothesis (H1) represents what the researcher seeks to find evidence for. They form the basis of statistical testing by clearly defining what is being tested and directing the type of statistical analysis used.

One-Tailed vs Two-Tailed Tests

A one-tailed test examines deviations from the hypothesized parameter in a single direction, whereas a two-tailed test looks for deviations in both directions. The choice between a one-tailed and a two-tailed test depends on the research question and the directionality of the alternative hypothesis. This decision affects the critical region and the interpretation of both the decision rule and the p-value.

Significance Level and Decision Rule

The significance level, often denoted as alpha, is the pre-determined threshold for how unlikely an observation must be under the null hypothesis to justify rejecting it. The decision rule identifies the critical region of the test statistic; if the computed statistic falls in this region, the null hypothesis is rejected. This rule is essential as it controls the probability of committing a Type I error.

p-value

The p-value quantifies the probability of observing a test statistic as extreme or more extreme than the one calculated from the sample data, assuming that the null hypothesis is true. It provides a measure of the strength of evidence against H0. A small p-value, relative to the significance level, indicates that the observed data are unlikely under the null hypothesis and supports the rejection of H0.

Transcript

00:01 Right, so here we have small observations from a normal population.

00:06 Samplemean is given, the populations, the derivation is given.

00:11 And we're going to conduct the test hypothesis using 0 .025 significance level.

00:19 So here is our hypotheses.

00:22 So this is a one -tailed test because we're looking as the is the mean, population mean less than this stated.

00:31 Stated population.

00:35 And the decision rule, so it's, we want to see is the mu less than a given value.

00:40 So our decision rule is going to be, based on the z score we get looking up this, the p value, it's a 0 .125.

00:56 So it's negative 1 .95 -9963.

01:01 So the decision rule is that, is the z score less than this score? critical value.

01:09 Let's figure that out.

01:11 So we take the mean, the sample mean of 215 minus the population mean divided by the deviation of 15 divided by the square root the sample size.

01:26 8.

01:38 Yeah, that's what it is.

01:45 And so our decision rule, because we just wanted to know if the is the mean less than 220, oh, okay, that was weird.

01:57 I thought that was weird.

01:58 When i did this before, i got a negative.

02:00 How did i get that? it should be 215 minus 220.

02:04 There it is.

02:05 The important thing is to find, not only find your mistake, but go how to fix them.

02:08 There is the z score...