Given the following hypotheses: H0: μ=100 H1: μ≠ 100 A random sample of six resulted

Given the following hypotheses: H0: μ=100 H1: μ≠...

Given the following hypotheses: $$\begin{array}{l}H_{0}: \mu=100 \\H_{1}: \mu \neq 100\end{array}$$ A random sample of six resulted in the following values: $118,105,112,119,105,$ and 111 . Assume a normal population. Using the .05 significance level, can we conclude the mean is different from $100 ?$ a. State the decision rule. b. Compute the value of the test statistic. c. What is your decision regarding the null hypothesis? d. Estimate the $p$ -value.

Given the following hypotheses:
$$\begin{array}{l}H_{0}: \mu=100 \\H_{1}: \mu \neq 100\end{array}$$
A random sample of six resulted in the following values: $118,105,112,119,105,$ and 111 . Assume a normal population. Using the .05 significance level, can we conclude the mean is different from $100 ?$
a. State the decision rule.
b. Compute the value of the test statistic.
c. What is your decision regarding the null hypothesis?
d. Estimate the $p$ -value.

Key Concepts

p-Value

The p-value is the probability, under the assumption that the null hypothesis is true, of obtaining a test statistic as extreme as or more extreme than the one observed. It quantifies the evidence against the null hypothesis, with a smaller p-value indicating stronger evidence to reject H?.

Decision Rule

The decision rule in hypothesis testing is a guideline formulated before data analysis, which specifies the criteria for rejecting or failing to reject the null hypothesis. It is typically based on whether the computed test statistic falls into the critical region defined by the significance level.

t-Distribution

The t-distribution is a probability distribution used in hypothesis testing when the sample size is small and the population standard deviation is unknown. It is similar to the normal distribution but has heavier tails, and the exact shape of the distribution depends on the degrees of freedom.

Degrees of Freedom

Degrees of freedom refer to the number of independent values in a calculation that are free to vary. In the context of a one-sample t-test, the degrees of freedom are typically calculated as the sample size minus one, and they affect the shape of the t-distribution used for determining critical values.

Significance Level

The significance level (often denoted by ?) is the probability threshold set by the researcher for rejecting the null hypothesis when it is true (Type I error). Commonly used values are 0.05 or 0.01, and it defines the critical region in the sampling distribution where the null hypothesis will be rejected.

Test Statistic

A test statistic is a standardized value calculated from sample data during a hypothesis test. It is used to determine whether to reject the null hypothesis by comparing it to a critical value from a relevant distribution. Its calculation often involves the sample mean, sample standard deviation, and sample size.

Null and Alternative Hypotheses

In hypothesis testing, the null hypothesis (H?) usually states that there is no significant difference or effect, while the alternative hypothesis (H?) suggests that there is a difference. In the context of testing a mean, H? specifies a particular population mean, and H? proposes that the population mean differs from that specified value.

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 formulating a null hypothesis (no effect or status quo) and an alternative hypothesis (the effect or difference that one suspects), then using a test statistic to decide if the observed data is extreme enough to reject the null hypothesis.

Transcript

00:01 All right, so we're given the following hypotheses that mu is 100 for the null and mu is not 100 for the alternative.

00:07 Excuse me, that the mu is not 100 for the alternative.

00:11 And we're given six samples here and we're assuming it's a normal population.

00:16 And we're going to know at the 0 .05 significance level, if we can conclude, the mean is different from 100.

00:23 So let's state the decision rule.

00:26 So we need the critical value.

00:27 So it's a two -tailed test.

00:30 So we're going to go to our t -table and look up five degrees of freedom.

00:39 So five degrees of freedom at the 05.

00:45 Where we go? 0 .05 here.

00:49 There we go.

00:49 Two -tailed test.

00:50 0 .05.

00:52 You could also get there by going 0 .05 for one -tailed, divide by 2 .025 for two -tale tests.

00:59 That's how we can get it.

01:00 So there's our value.

01:03 2 .571.

01:04 We also say it could be the negative of it, or sorry, we're looking for, is it greater than this value of 2 .5 or less than the negative 2 .5? and let's compute our statistic.

01:16 Oh, and it's already done for us...

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