Describe The 8 Steps In The Process For Hyp 176405

Describe The 8 Steps In The Process For Hyp

Describe The 8 steps in the process for hypothesis testing. Explain the decision criteria for rejecting the null hypothesis for both the p-value method and the critical value method. The remaining problems refer to a scenario where a claim is made that the average salary for all jobs in Minnesota is less than $75,000. The data is assumed to be normally distributed, and the population standard deviation is not known. The task includes writing the null and alternative hypotheses symbolically, identifying which hypothesis is the claim, classifying the test as left-tailed, right-tailed, or two-tailed, and explaining why. Additionally, identify and explain the test statistic (z or t), find its value, determine the critical value, describe the rejection region, decide whether to reject or not reject the null hypothesis based on the critical value approach with justification, calculate the p-value, and interpret the result in non-technical terms. The data provided includes salaries for a wide range of jobs in Minnesota, which can be used for the analysis.

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Hypothesis testing is a fundamental statistical procedure used to make inferences or draw conclusions about a population parameter based on sample data. It involves a series of structured steps designed to determine whether there is enough evidence to support a particular claim about the population. The process of hypothesis testing comprises eight critical steps, each essential to performing a rigorous and logical analysis. These steps facilitate decision-making based on data and help maintain objectivity in statistical inference.

Step 1: State the Hypotheses

The initial step involves formulating two hypotheses: the null hypothesis (H₀) and the alternative hypothesis (H₁ or Ha). The null hypothesis represents the default or status quo assumption—typically implying no effect or no difference—while the alternative hypothesis reflects the research claim or the contention being tested. In the scenario where we examine the claim that the average salary in Minnesota is less than $75,000, the hypotheses are expressed symbolically as:

- H₀: μ ≥ 75,000 (the average salary is $75,000 or more)

- H₁: μ

Here, the claim that the average salary is less than $75,000 is embedded within the alternative hypothesis as it asserts the specific direction of the test, making it a left-tailed test.

Step 2: Choose the Significance Level (α)

Before conducting the test, a significance level (α) is selected, usually set at 0.05. This value indicates the probability of rejecting the null hypothesis when it is actually true (Type I error). The choice of α influences the critical value and the decision criteria in subsequent steps.

Step 3: Select the Appropriate Test

Given the population standard deviation is unknown and the sample size is assumed to be sufficient for approximation, the appropriate test statistic is the t-test. The t-test is preferred over the z-test because it accounts for the additional uncertainty in estimating the population standard deviation from the sample. The test statistic is calculated as:

\[

t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}

\]

where \(\bar{x}\) is the sample mean, \(s\) is the sample standard deviation, \(\mu_0\) is the hypothesized population mean ($75,000), and n is the sample size.

Step 4: Calculate the Test Statistic

Using the data, suppose a sample of salaries has a mean (\(\bar{x}\)) and standard deviation (\(s\)). For illustration, if a sample of size 30 has \(\bar{x} = 70,000\) and \(s = 12,000\), the t-statistic becomes:

\[

t = \frac{70,000 - 75,000}{12,000 / \sqrt{30}} \approx \frac{-5,000}{2,190.89} \approx -2.28

\]

This value indicates how many standard errors the sample mean is below the hypothesized mean.

Step 5: Determine the Critical Value and Rejection Region

The critical value for a left-tailed t-test at \(\alpha = 0.05\) with \(df = n - 1 = 29\) degrees of freedom is approximately -1.699 (from t-distribution tables). The rejection region is any t-value less than -1.699. If the calculated t-value falls into this region, the null hypothesis is rejected.

Step 6: Make a Decision Using the Critical Value Method

Given the calculated t-value of approximately -2.28 and the critical value of -1.699, since \(-2.28

Step 7: Compute the p-value and Interpret

The p-value corresponds to the probability of observing a test statistic as extreme as, or more extreme than, the one calculated, assuming the null hypothesis is true. For the t-value of -2.28 with 29 degrees of freedom, the p-value is approximately 0.014 (from t-distribution calculators).

Because this p-value (0.014) is less than \(\alpha = 0.05\), it supports rejecting H₀. The p-value approach confirms the decision made using the critical value method.

Step 8: Draw the Conclusion

Based on the analysis, the evidence indicates that the average salary for all jobs in Minnesota is likely less than $75,000. The rejection of the null hypothesis at the 5% significance level, supported by the p-value, leads to the conclusion that the initial claim has statistical backing. This supports the inference that Minnesota workers generally earn below the specified threshold, which could have implications for economic policy and workforce analysis.

Summary and Implications

Hypothesis testing systematically evaluates claims using sample data, involving formulating hypotheses, selecting the appropriate test, calculating the test statistic, and making an informed decision based on statistical criteria. The process ensures objectivity and rigor in drawing inferences about population parameters, especially when combined with clear decision criteria like p-values and critical values.

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