Consider the following hypothesis test. He: μ1-μ2=0 H2: μ1-μ2 ≠0 The following resul

step 1 Consider the hypothesis test. The following results are for two independent samples taken from t

Consider the following hypothesis test. He: μ1-μ2=0 ...

Consider the following hypothesis test. $$ \begin{aligned} & H_e: \mu_1-\mu_2=0 \\ & H_2: \mu_1-\mu_2 \neq 0 \end{aligned} $$ The following results are for two independent samples taken from the two populations. $$ \begin{array}{ll} \text { Sample 1 } & \text { Sample 2 } \\ n_1=80 & n_2=70 \\ \bar{x}_1=104 & \bar{x}_3=106 \\ \sigma_1=8.4 & \sigma_2=7.6 \end{array} $$ a. What is the value of the test statistic? b. What is the $p$-value? c. With $\alpha=.05$, what is your hypothesis testing conclusion?

Consider the following hypothesis test.
$$
\begin{aligned}
& H_e: \mu_1-\mu_2=0 \\
& H_2: \mu_1-\mu_2 \neq 0
\end{aligned}
$$
The following results are for two independent samples taken from the two populations.
$$
\begin{array}{ll}
\text { Sample 1 } & \text { Sample 2 } \\
n_1=80 & n_2=70 \\
\bar{x}_1=104 & \bar{x}_3=106 \\
\sigma_1=8.4 & \sigma_2=7.6
\end{array}
$$
a. What is the value of the test statistic?
b. What is the $p$-value?
c. With $\alpha=.05$, what is your hypothesis testing conclusion?

Key Concepts

Hypothesis Testing

This is a statistical decision-making process where we begin by formulating a null hypothesis, typically representing no effect or difference, and an alternative hypothesis representing the presence of an effect or difference. The goal is to use sample data to decide whether there is enough evidence to reject the null hypothesis in favor of the alternative hypothesis.

Two-Sample Z-Test

A two-sample z-test is used to compare the means of two independent populations when the population standard deviations are known. It involves computing a test statistic based on the difference between the sample means, the known standard deviations, and the sample sizes, assuming that the sampling distributions are approximately normal.

Test Statistic

The test statistic quantifies how far the observed sample statistic is from what is expected under the null hypothesis, measured in standard error units. In the context of a two-sample z-test, it is calculated by subtracting the hypothesized difference (usually zero) from the observed difference between sample means and then dividing by the standard error of the difference.

p-value

The p-value represents the probability, under the null hypothesis, of obtaining a result as extreme as, or more extreme than, the observed test statistic. It is used to measure the strength of evidence against the null hypothesis; a smaller p-value indicates stronger evidence to reject the null hypothesis.

Significance Level and Hypothesis Conclusion

The significance level, denoted by alpha (?), is a predetermined threshold for rejecting the null hypothesis. If the p-value is less than or equal to ?, the null hypothesis is rejected, indicating that the observed data is statistically significant. If the p-value exceeds ?, there is insufficient evidence to reject the null hypothesis.

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