What Critical Region For Z To Test The Hypothesis
What critical region(s) for z would be used to test the hypotheses: versus , for = 0.05 ? and State the null hypothesis Ho and the alternative hypothesis Ha that would be used for a hypothesis test of the following
Evaluate the critical regions for z in hypothesis testing at a significance level of 0.05, and formulate the null and alternative hypotheses for testing whether the mean age of students enrolled in evening classes at a certain college is greater than 26 years.
Paper For Above instruction
Hypothesis testing is a fundamental aspect of inferential statistics, enabling researchers to make decisions about population parameters based on sample data. The first component of a hypothesis test involves establishing null and alternative hypotheses, followed by determining the critical region(s) that define the boundaries for rejecting the null hypothesis. This process is essential in evaluating claims such as whether the mean age of students exceeds a specific value.
To address the first part of the question concerning critical regions for the z-test, it is important to understand the context of significance levels. The significance level, denoted as α, typically indicates the probability of rejecting the null hypothesis when it is true (Type I error). For a significance level of α = 0.05, the critical regions depend on the nature of the test—whether it is one-tailed or two-tailed.
If the hypothesis test is designed to determine whether the true mean age is greater than 26 years, a right-tailed test would be appropriate. The null hypothesis (Ho) would state that there is no difference or that the mean age is less than or equal to 26, while the alternative hypothesis (Ha) would indicate that the mean age is greater than 26:
Null Hypothesis (Ho):
μ ≤ 26
Alternative Hypothesis (Ha):
μ > 26
In the context of a z-test, the test statistic z follows a standard normal distribution under the null hypothesis. The critical value corresponding to α = 0.05 for a right-tailed test can be found in z-tables or statistical software, which yields approximately:
- zcritical ≈ 1.645
This means that the critical region (the rejection region) consists of all values of z greater than 1.645. If the computed z-statistic exceeds this value, the null hypothesis can be rejected with 95% confidence, indicating sufficient evidence that the mean age is greater than 26 years.
Conversely, for a two-tailed test at the same significance level, the critical regions would be at both ends of the distribution, with z-values approximately ±1.96. In such a case, the null hypothesis would be:
Null Hypothesis (Ho):
μ = 26
Alternative Hypothesis (Ha):
μ ≠ 26
And the critical regions would be z 1.96, corresponding to the upper and lower tails of the standard normal distribution, capturing 2.5% in each tail, totaling a 5% significance level.
In conclusion, the critical region for the z-test at a 0.05 significance level when testing if the mean exceeds 26 is z > 1.645. The null and alternative hypotheses framing this test are as specified above, reflecting the research question concerning the mean age of students in evening classes.
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