H0:p=0.5
H1:p>0.5

Your sample consists of 109 subjects, with 49 successes. Calculate the test statistic, rounded to 2 decimal places

z=

To calculate the test statistic for the given hypotheses, we need to use the formula for the z-test for proportions. The hypotheses are:

- \( H_0: p = 0.5 \)
- \( H_1: p > 0.5 \)

Given:
- \( n = 109 \) (total number of subjects)
- \( x = 49 \) (number of successes)
- \( p_0 = 0.5 \) (proportion under the null hypothesis)

The sample proportion \( \hat{p} \) is calculated as:
\[ \hat{p} = \frac{x}{n} = \frac{49}{109} \approx 0.4495 \]

The standard error of the proportion under the null hypothesis is:
\[ \text{SE} = \sqrt{\frac{p_0 (1 - p_0)}{n}} = \sqrt{\frac{0.5 \cdot 0.5}{109}} = \sqrt{\frac{0.25}{109}} \approx \sqrt{0.00229} \approx 0.0478 \]

The test statistic \( z \) is calculated as:
\[ z = \frac{\hat{p} - p_0}{\text{SE}} = \frac{0.4495 - 0.5}{0.0478} \approx \frac{-0.0505}{0.0478} \approx -1.06 \]

So, the test statistic rounded to two decimal places is:
\[ z \approx -1.06 \]

Would you like more details or have any questions about the steps? Here are some related questions to consider:

1. What is the purpose of a z-test for proportions?
2. How do we interpret the test statistic in the context of hypothesis testing?
3. What are the assumptions underlying the z-test for proportions?
4. How is the standard error calculated for a proportion?
5. What is the significance level and how does it relate to the test statistic?
6. How do we determine the p-value associated with the test statistic?
7. What would be the conclusion if the test statistic were significantly positive?
8. What are some common applications of the z-test for proportions?

**Tip:** When conducting hypothesis tests, ensure that the sample size is sufficiently large to justify the use of the normal approximation.

Learn how to calculate the Z-test statistic for proportions with hypotheses p = 0.5 and p > 0.5. This problem involves determining the test statistic using sample data of 109 subjects, with 49 successes. Detailed step-by-step solution provided.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation