Math Problem Statement
Solution
Let’s solve the hypothesis test step by step:
Hypotheses:
- (right-tailed test)
Given Data:
- Sample size
- Number of successes
- Significance level
Step 1: Sample Proportion
Calculate the sample proportion:
Part a: Test Statistic
The test statistic for a proportion is calculated as:
Substitute the values:
First, calculate the denominator:
Now calculate :
So, the test statistic is:
Part b: P-value
For a right-tailed test, the p-value is the area to the right of on the standard normal distribution.
Using a Z-table or technology:
The p-value is:
Part c: Compare P-value and
Since , the p-value is less than .
Part d: Decision
Since , we reject the null hypothesis.
Part e: Conclusion
The conclusion is:
If you need any further explanation or detailed breakdown of any step, let me know!
5 Related Questions:
- How is the sample size used to affect the test statistic in a hypothesis test for proportions?
- What would happen if the significance level were lower (e.g., )?
- How does a right-tailed test differ from a two-tailed test in hypothesis testing?
- Can a small sample size still yield significant results? Why or why not?
- How do you interpret the p-value in the context of hypothesis testing?
Tip:
Always double-check calculations for the test statistic, especially the denominator, as small errors can affect the final conclusion!
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Hypothesis Testing
Proportions
Right-tailed Tests
Significance Levels
Formulas
z = (p̂ - p0) / √[p0(1 - p0) / n]
p̂ = x / n
Theorems
Central Limit Theorem
Standard Normal Distribution
Suitable Grade Level
Grades 11-12, College Statistics