Describe The 8 Steps In The Process For Hyp

Describe The 8 Steps In The Process For Hyp

Deliberate the 8 steps in the process for hypothesis testing. Explain the decision criteria for rejecting the null hypothesis utilizing both the p-value method and the critical value method. Provide detailed, step-by-step explanations and solutions based on a scenario where a claim asserts that the average salary for all jobs in Minnesota is less than $65,000. Assume data is normally distributed with an unknown population standard deviation, and analyze the data accordingly to test this claim.

Paper For Above instruction

Hypothesis testing is a fundamental statistical procedure used to make inferences or draw conclusions about a population parameter based on sample data. The process involves several methodical steps, each critical to ensuring accurate and reliable results. In this paper, we explore the eight key steps involved in hypothesis testing, elucidate the decision criteria for rejecting the null hypothesis through both the p-value method and the critical value approach, and apply these to a specific scenario involving salary data in Minnesota.

Step 1: State the Hypotheses

The initial step involves formulating the null hypothesis (H0) and the alternative hypothesis (Ha). These hypotheses are precise statements about the parameter in question. In our scenario, the claim is that the average salary in Minnesota is less than $65,000. Therefore:

- Null hypothesis (H0): μ ≥ 65,000

- Alternative hypothesis (Ha): μ

The null hypothesis reflects no effect or status quo, while the alternative aligns with the claim, making this a left-tailed test because we're testing whether the mean salary is significantly less than $65,000.

Step 2: Set the Significance Level (α)

The significance level, α, is a threshold for determining whether the observed data provides sufficient evidence against H0. Commonly set at 0.05, α quantifies the probability of a Type I error—incorrectly rejecting H0 when it is true. Choosing α involves balancing the risk of false positives with the power of the test.

Step 3: Select the Appropriate Test Statistic

Given the data is normally distributed, but the population standard deviation is unknown, we employ the t-test for the mean. The test statistic is calculated as:

t = (x̄ - μ0) / (s / √n)

where x̄ is the sample mean, μ0 is the population mean under H0 (here, 65,000), s is the sample standard deviation, and n is the sample size. This statistic assesses how far the sample mean deviates from the hypothesized mean in units of standard error.

Step 4: Calculate the Test Statistic

Using the salary data, suppose we have a sample mean salary of x̄ = $64,150 and an estimated sample standard deviation s, based on the provided data, and a particular sample size n. Calculating the test statistic involves plugging these values into the formula. For example, assuming n=50, and s=15,000:

t = (64,150 - 65,000) / (15,000 / √50)

t = (-850) / (15,000 / 7.071)

t = (-850) / 2,121.

t ≈ -0.4

This value will then be used for the decision-making process, with actual data leading to a precise t-value.

Step 5: Determine the Critical Value and Rejection Region

To establish the rejection criterion, consult t-distribution tables at the chosen α level with n-1 degrees of freedom. For α=0.05 and df=49, the critical value for a left-tailed test is approximately -1.68. The rejection region comprises all t-values less than -1.68. Since our calculated t-value of about -0.4 does not enter this region, we would not reject H0 based on the critical value approach.

Step 6: Decision Using Critical Value Method and Explanation

Because the calculated t-value is not less than the critical value (-0.4 > -1.68), we do not reject H0. This suggests insufficient evidence to support the claim that the average salary is less than $65,000. In simple terms, based on our sample data, we cannot conclude that the average salary in Minnesota falls below $65,000; the observed difference could be due to random variation.

Step 7: Calculate the P-Value and Make a Decision

The p-value indicates the probability of observing a sample mean as extreme as x̄, or more, under H0. Using the t-distribution with the computed t-value of -0.4, the p-value corresponds to the area to the left of t = -0.4. Looking this up in t-tables or using statistical software, the p-value is approximately 0.34.

Since p-value (0.34) > α (0.05), the evidence is not strong enough to reject H0. This aligns with the decision from the critical value method, indicating consistency between the two approaches. In plain language, there’s a 34% chance of seeing such a sample mean if the true average salary is $65,000 or higher—far above our threshold for significance.

Step 8: Conclusion

Based on the analysis, we conclude there is insufficient statistical evidence to support the claim that the average salary in Minnesota is less than $65,000. The data does not suggest a statistically significant difference from the hypothesized mean. Therefore, the original claim cannot be accepted based on this sample.

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