Consider the following hypothesis test: H0: μ≥80 Ha: μ<80 A sample of 100 is used an

step 1 Okay, so here in part A, we have the size of our sample N is 100. The sample mean, X bar, is 78

Consider the following hypothesis test: H0: μ≥80 ...

Consider the following hypothesis test: $$ \begin{aligned} & H_0: \mu \geq 80 \\ & H_{\mathrm{a}}: \mu<80 \end{aligned} $$ A sample of 100 is used and the population standard deviation is 12 . Compute the p-value and state your conclusion for each of the following sample results. Use $a=.01$. a. $\bar{x}=78.5$ b. $\bar{x}=77$ c. $\vec{x}=75.5$ d. $\tilde{\boldsymbol{x}}=81$

Consider the following hypothesis test:
$$
\begin{aligned}
& H_0: \mu \geq 80 \\
& H_{\mathrm{a}}: \mu<80
\end{aligned}
$$
A sample of 100 is used and the population standard deviation is 12 . Compute the p-value and state your conclusion for each of the following sample results. Use $a=.01$.
a. $\bar{x}=78.5$
b. $\bar{x}=77$
c. $\vec{x}=75.5$
d. $\tilde{\boldsymbol{x}}=81$

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical framework used to decide whether there is enough evidence in a sample of data to infer that a certain condition holds for the entire population. It involves comparing a test statistic, calculated from sample data, to a theoretical distribution under the null hypothesis.

Null and Alternative Hypotheses

The null hypothesis (H?) is a statement about a population parameter that indicates no effect or no difference; it serves as a baseline assumption. In contrast, the alternative hypothesis (Ha) represents a new claim or effect that the researcher wants to test, and it defines the direction or nature of the deviation from the null hypothesis.

Significance Level (?)

The significance level, denoted by alpha (?), is the threshold probability used to decide whether to reject the null hypothesis. It quantifies the acceptable risk of making a Type I error, which is the error of wrongly rejecting a true null hypothesis. Common values for ? are 0.05 and 0.01.

p-value

The p-value is the probability of obtaining a test statistic at least as extreme as the one computed from the sample data, assuming the null hypothesis is true. It provides a measure of the strength of evidence against the null hypothesis, with a lower p-value indicating stronger evidence for the alternative hypothesis.

One-sample z-test

A one-sample z-test is used to assess whether the mean of a sample is statistically significantly different from a known or hypothesized population mean when the population standard deviation is known. This test is applicable when the sample size is large enough for the central limit theorem to justify the use of the normal distribution.

One-tailed Test

A one-tailed test is a hypothesis test where the critical region for rejecting the null hypothesis is located entirely in one tail of the probability distribution. It is used when the research hypothesis specifies a direction of the effect, such as testing whether the mean is less than or greater than a certain value.

Transcript

00:01 Okay, so here in part a, we have the size of our sample n is 100.

00:04 The sample mean, x bar, is 78 .5, and our population's standard deviation, sigma, is equal to 12.

00:10 So the hypotheses to be tested are h -0, the null hypothesis, is mu is greater than equal to 80, and our alternative hypothesis is that mu is less than 80.

00:21 So then we get the test statistic for the hypotheses about a population mean.

00:26 That's going to be z is equal to here, x bar, minus mu or minus um new not divided by sigma divided by the square of n so that's going to give us negative 1 .25 and then the area under the standard normal curve to the left of z equals negative 1 .25 is obtained from the standard normal tables and we get that's going to be equal um so we get that the probability of z being less than or equal to negative 1 .25 is going to be equal to 0 .106.

00:59 So here's our p value for this left tail test is 0 .06.

01:03 The given level of significance is alpha is equal to 0 .01.

01:09 And we have that 0 .106 is greater than 0 .01.

01:13 So therefore, we fail to reject the null hypothesis because our p value here is greater than alpha.

01:25 And then for part b, the size of the sample is 100.

01:29 Our sample mean.

01:31 So again, n is 100.

01:33 Sample mean x bar is 77.

01:36 And we get that sigma is equal to 12.

01:39 So the hypotheses now, again, h not, is that mu is going to be greater than equal to 80.

01:46 And the alternative hypothesis is that mu is going to be less than 80...