Consider the following hypothesis test: H0: μ≥20 Ha: μ<20 A sample of 50 provided a

step 1 Hello, so here we're given the sample size n is 50. The sample mean X bar is 19 .4, and the popu

Consider the following hypothesis test: H0: μ≥20 ...

Consider the following hypothesis test: $$ \begin{aligned} & H_0: \mu \geq 20 \\ & H_{\mathrm{a}}: \mu<20 \end{aligned} $$ A sample of 50 provided a sample mean of 19.4 . The population standard deviation is 2 . a. Compute the value of the test statistic. b. Whar is the p-value? c. Using $a=.05$, what is your conclusion? d. Whar is the rejection rule using the critical value? What is your conclusion?

Consider the following hypothesis test:
$$
\begin{aligned}
& H_0: \mu \geq 20 \\
& H_{\mathrm{a}}: \mu<20
\end{aligned}
$$
A sample of 50 provided a sample mean of 19.4 . The population standard deviation is 2 .
a. Compute the value of the test statistic.
b. Whar is the p-value?
c. Using $a=.05$, what is your conclusion?
d. Whar is the rejection rule using the critical value? What is your conclusion?

Key Concepts

Significance Level (?)

The significance level, denoted by ?, is a threshold chosen by the researcher to decide when to reject the null hypothesis. It represents the probability of committing a Type I error, which is rejecting the null hypothesis when it is actually true. Common choices for ? are 0.05, 0.01, or 0.10.

P-value

The p-value is the probability, under the null hypothesis, of obtaining a test statistic at least as extreme as the one computed from the sample data. It provides a measure of the strength of the evidence against the null hypothesis, with smaller p-values indicating stronger evidence in its rejection.

Rejection Region and Critical Value

The rejection region is a set of values for the test statistic that leads to the rejection of the null hypothesis. The critical value is the boundary value that separates the rejection region from the non-rejection region. When the test statistic falls into the rejection region based on the chosen significance level, the null hypothesis is rejected in favor of the alternative hypothesis.

One-Sample z-Test

A one-sample z-test is used to determine whether a sample mean significantly differs from a known or hypothesized population mean when the population standard deviation is known. This test relies on the standard normal distribution to compute the z-score, which measures the number of standard errors the sample mean is from the population mean.

Null and Alternative Hypotheses

In hypothesis testing, the null hypothesis (H0) represents a default position or a statement of no effect or no difference, while the alternative hypothesis (Ha) represents the claim being tested against H0. The setup of these hypotheses determines whether a test is one-tailed or two-tailed and directs the focus of the statistical analysis.

Test Statistic

The test statistic in a one-sample z-test quantifies the deviation of the sample mean from the population mean in units of the standard error. It is calculated by subtracting the population mean from the sample mean and then dividing by the standard error, which is the population standard deviation divided by the square root of the sample size.

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