Math Problem Statement

You wish to test the following claim at a significance level of α = 0.05. H0: p = 0.62; Ha: p > 0.62. You obtain a sample of size n = 491 in which there are 337 successful observations. Calculate the test statistic, p-value, and make a conclusion.

Solution

Let’s solve the hypothesis test step by step:


Hypotheses:

  • H0:p=0.62H_0: p = 0.62
  • Ha:p>0.62H_a: p > 0.62 (right-tailed test)

Given Data:

  • Sample size n=491n = 491
  • Number of successes x=337x = 337
  • Significance level α=0.05\alpha = 0.05

Step 1: Sample Proportion

Calculate the sample proportion: p^=xn=3374910.686\hat{p} = \frac{x}{n} = \frac{337}{491} \approx 0.686


Part a: Test Statistic

The test statistic for a proportion is calculated as: z=p^p0p0(1p0)nz = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}}

Substitute the values:

  • p^=0.686\hat{p} = 0.686
  • p0=0.62p_0 = 0.62
  • n=491n = 491

z=0.6860.620.62(10.62)491z = \frac{0.686 - 0.62}{\sqrt{\frac{0.62(1 - 0.62)}{491}}}

First, calculate the denominator: 0.62(0.38)491=0.2356491=0.000480.0219\sqrt{\frac{0.62(0.38)}{491}} = \sqrt{\frac{0.2356}{491}} = \sqrt{0.00048} \approx 0.0219

Now calculate zz: z=0.6860.620.0219=0.0660.02193.014z = \frac{0.686 - 0.62}{0.0219} = \frac{0.066}{0.0219} \approx 3.014

So, the test statistic is: z=3.014z = 3.014


Part b: P-value

For a right-tailed test, the p-value is the area to the right of z=3.014z = 3.014 on the standard normal distribution.

Using a Z-table or technology: P(z>3.014)0.0013P(z > 3.014) \approx 0.0013

The p-value is: P-value=0.0013\text{P-value} = 0.0013


Part c: Compare P-value and α\alpha

  • P-value=0.0013\text{P-value} = 0.0013
  • α=0.05\alpha = 0.05

Since P-value<α\text{P-value} < \alpha, the p-value is less than α\alpha.


Part d: Decision

Since P-value<α\text{P-value} < \alpha, we reject the null hypothesis.


Part e: Conclusion

The conclusion is: There is sufficient evidence to warrant rejection of the claim that the population proportion is greater than 0.62.\text{There is sufficient evidence to warrant rejection of the claim that the population proportion is greater than 0.62.}


If you need any further explanation or detailed breakdown of any step, let me know!


5 Related Questions:

  1. How is the sample size used to affect the test statistic in a hypothesis test for proportions?
  2. What would happen if the significance level α\alpha were lower (e.g., α=0.01\alpha = 0.01)?
  3. How does a right-tailed test differ from a two-tailed test in hypothesis testing?
  4. Can a small sample size still yield significant results? Why or why not?
  5. 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!

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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