Consider the following hypothesis test: Ho: μ=22 Ha: μ≠22 A sample of 75 is used and

step 1 Hello, so here we're given that alpha is equal to 0 .01. We have that our sample n is 75, our sa

Consider the following hypothesis test: Ho: μ=22 ...

Consider the following hypothesis test: $$ \begin{aligned} & H_{\mathrm{o}}: \mu=22 \\ & H_{\mathrm{a}}: \mu \neq 22 \end{aligned} $$ A sample of 75 is used and the population standard deviation is 10. Compute the $p$-value and state your conclusion for each of the following sample results. Use $\alpha=.01$. a. $\bar{x}=23$ b. $\bar{x}=25,1$ c. $\tilde{x}=20$

Consider the following hypothesis test:
$$
\begin{aligned}
& H_{\mathrm{o}}: \mu=22 \\
& H_{\mathrm{a}}: \mu \neq 22
\end{aligned}
$$
A sample of 75 is used and the population standard deviation is 10. Compute the $p$-value and state your conclusion for each of the following sample results. Use $\alpha=.01$.
a. $\bar{x}=23$
b. $\bar{x}=25,1$
c. $\tilde{x}=20$

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to decide whether there is enough evidence in a sample to infer that a certain condition holds for the entire population. It involves setting up a null hypothesis (a statement of no effect or no difference) and an alternative hypothesis, then collecting sample data to assess whether the observed data are consistent with the null hypothesis or support the alternative hypothesis.

Z-Test

A Z-test is employed when the population standard deviation is known and either the sample size is large or the underlying distribution is normal. It involves computing a test statistic using the sample mean, the hypothesized population mean, the known standard deviation, and the sample size, then comparing this statistic to a standard normal distribution to determine the likelihood of observing data as extreme as the actual sample under the null hypothesis.

Standard Error

The standard error quantifies the variability of the sampling distribution of the sample mean and is computed as the population standard deviation divided by the square root of the sample size. It is fundamental in standardizing the difference between the observed sample mean and the hypothesized population mean when performing a Z-test.

P-Value Calculation

The p-value is the probability of obtaining a test statistic at least as extreme as the one observed, assuming the null hypothesis is true. In a two-tailed test, this probability is calculated for both tails of the distribution. A smaller p-value indicates stronger evidence against the null hypothesis, and it is compared with a predetermined significance level to make inferences.

Significance Level and Decision Making

The significance level, denoted by alpha, is the threshold at which the null hypothesis is rejected. It represents the probability of making a Type I error – incorrectly rejecting the null hypothesis when it is actually true. After calculating the p-value, one compares it to alpha; if the p-value does not exceed alpha, there is sufficient evidence to reject the null hypothesis.

Transcript

00:01 Hello, so here we're given that alpha is equal to 0 .01.

00:03 We have that our sample n is 75, our sample mean, x bar is 23, and our population standard deviation is sigma is equal to 10.

00:12 So here we have our hypotheses are that h -not.

00:14 We have that mu is equal to 22, and our alternative hypothesis is that mu is not equal to 22.

00:20 So we then have our test statistic is given by x bar minus mu, which is 23 minus 22 divided by sigma divided by the square of n.

00:33 So that's 10 divided by the square root of 75, which is going to give us 0 .87.

00:40 So then we get the area under the standard normal curve to the right of z equals 0 .87 is obtained from the standard normal table, and we get that's going to be equal to 1 minus 0 .08078, which is 0 .19.

00:57 2 .2.

00:58 So then we can double this to find the p value for the two -tail test.

01:02 So therefore, we get that, whoops, we get that p is going to be equal to two times 0 .1922.

01:10 So therefore, that's equal to 0 .3844, which we have is greater than 0 .01.

01:20 So therefore, since our p value here is greater than alpha, we fail to reject um, h -not, the null hypothesis at a 1 % level of, oops, at, well, at a 1 % level of significance.

01:41 And then for part b, we have that alpha is 0 .01.

01:46 Samples again is 75 and the sample mean now is 25 .1 with our standard deviation being 10.

01:53 Um, so again, we have our hypotheses that mu is equal to 22 and our alternatives that mu is not equal to 22...

Please give Ace some feedback