Consider the following hypothesis test: Ho: μ≥45 Ha: μ<45 A sample of 36 is used. Id

step 1 Okay, so here in part A, we're given the sample n is 36. So the degrees of freedom is 36 minus 1

Consider the following hypothesis test: Ho: μ≥45 ...

Consider the following hypothesis test: $$ \begin{aligned} & H_{\mathrm{o}}: \mu \geq 45 \\ & H_{\mathrm{a}}: \mu<45 \end{aligned} $$ A sample of 36 is used. Identify the $p$-value and state your conclusion for each of the following sample results. Use $a=.01$. a. $\vec{x}=44$ and $s=5.2$ b. $\bar{x}=43$ and $s=4.6$ c. $\vec{x}=46$ and $s=5.0$

Consider the following hypothesis test:
$$
\begin{aligned}
& H_{\mathrm{o}}: \mu \geq 45 \\
& H_{\mathrm{a}}: \mu<45
\end{aligned}
$$
A sample of 36 is used. Identify the $p$-value and state your conclusion for each of the following sample results. Use $a=.01$.
a. $\vec{x}=44$ and $s=5.2$
b. $\bar{x}=43$ and $s=4.6$
c. $\vec{x}=46$ and $s=5.0$

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical process 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 null hypothesis, which represents a default or assumed state, against an alternative hypothesis that embodies a different outcome.

One-Sample t-Test

A one-sample t-test is used to determine whether the sample mean is statistically significantly different from a known or hypothesized population mean when the population standard deviation is unknown. This test calculates a t-statistic based on the sample mean, sample standard deviation, and sample size, and uses the t-distribution with appropriate degrees of freedom.

p-value

The p-value is a measure that helps determine the significance of the results from a hypothesis test. It represents the probability of obtaining a test statistic at least as extreme as the one observed, assuming the null hypothesis is true. A smaller p-value indicates stronger evidence against the null hypothesis.

Significance Level

The significance level, often denoted by alpha (?), is a threshold set by the researcher (in this case, 0.01) against which the p-value is compared. If the p-value is less than or equal to the significance level, the null hypothesis is rejected, indicating that the results are statistically significant.

Test Statistic

The test statistic in a one-sample t-test quantifies the difference between the sample mean and the hypothesized population mean relative to the variability in the data. This standardized value is used to determine the p-value and assess whether the observed deviation is statistically significant.

Transcript

00:01 Okay, so here in part a, we're given the sample n is 36.

00:04 So the degrees of freedom is 36 minus 1, which is 35.

00:08 The sample mean x bar is 44, and the sample standard deviation is 5 .2.

00:14 The hypotheses to be tested are h -0, which is mu greater than or equal to 45.

00:20 And then the alternative hypothesis would be mu is less than 45.

00:24 So here the test statistic is given by t is equal to x bar.

00:30 Minus mu not, that's going to be 44 minus 45, and then divided by s minus the square root of n, which is going to be 5 .2 divided by the square root of 36 or 6, which is going to be equal to negative 1 .15.

00:48 So then we get the area under the t distribution curve to the left of negative 1 .15 is the same as the area under the curve to the right of t equals 1 .15, which we get from the t table, and our degrees of freedom, which we then get that t is going to be equal 1 .15, is between 0 .852 and 1 .306.

01:11 So then the area in the upper tail roll, we see the area to the tail of the right of t equals 1 .15 is between 0 .2 and 0 .1, which we then get the p value must be between 0 .20 and 0 .10.

01:30 So given that alpha is 0 .05, we have that our p value is going to be greater than alpha.

01:39 So therefore, we then fail to reject the null hypothesis at a 1 % level of significance.

01:50 Then for part b, we're given the size of the sample n is 36.

01:56 We get our degrees of freedom then as 36 minus 1.

02:00 So that's 35.

02:02 And then our sample mean x bar is 43.

02:06 And our sample standard deviation is 4 .6.

02:12 So then the hypothesis to be tested again, h0, mu grade, then equal to 45, and the alternative, which be mu less than 45...