Find The Value Of The Test Statistic Z Z Z Using The Formula: Z = P ^ − P P Q N Z=\frac{\hat{p}-p}{\sqrt{\frac{pq}{n}}} Z = N Pq ​ ​ P ^ ​ − P ​ The Claim Is That The Proportion Of Accidental Deaths Of The Elderly Attributable To Residential Falls Is More Than 0.10. The Sample

Introduction

In statistics, hypothesis testing is a method used to determine whether there is enough evidence to reject a null hypothesis. In this article, we will focus on hypothesis testing for proportions, specifically using the formula $z=\frac{\hat{p}-p}{\sqrt{\frac{pq}{n}}}$ to find the value of the test statistic $z$. We will use a real-world example to illustrate the process.

Understanding the Formula

The formula $z=\frac{\hat{p}-p}{\sqrt{\frac{pq}{n}}}$ is used to calculate the test statistic $z$ for hypothesis testing of proportions. Here's a breakdown of the variables:

• $\hat{p}$: The sample proportion
• $p$: The population proportion (the null hypothesis value)
• $q$: The population proportion of the complement (i.e., $q = 1 - p$)
• $n$: The sample size
• $z$: The test statistic

The Null and Alternative Hypotheses

In this example, the claim is that the proportion of accidental deaths of the elderly attributable to residential falls is more than 0.10. This can be written as:

• Null hypothesis ($H_0$): $p \leq 0.10$
• Alternative hypothesis ($H_a$): $p > 0.10$

Calculating the Sample Proportion

To calculate the sample proportion, we need to know the number of successful outcomes (i.e., accidental deaths of the elderly attributable to residential falls) and the total sample size. Let's assume that we have a sample of 100 elderly individuals, and 15 of them died due to residential falls.

$\hat{p} = \frac{\text{Number of successful outcomes}}{\text{Total sample size}} = \frac{15}{100} = 0.15$

Calculating the Test Statistic $z$

Now that we have the sample proportion, we can plug in the values into the formula to calculate the test statistic $z$.

$z = \frac{\hat{p} - p}{\sqrt{\frac{pq}{n}}} = \frac{0.15 - 0.10}{\sqrt{\frac{0.10 \times 0.90}{100}}} = \frac{0.05}{\sqrt{0.0009}} = \frac{0.05}{0.0030} = 16.67$

Interpreting the Results

The calculated value of the test statistic $z$ is 16.67. This value indicates that the observed sample proportion is significantly higher than the null hypothesis value of 0.10. In other words, there is strong evidence to reject the null hypothesis and conclude that the proportion of accidental deaths of the elderly attributable to residential falls is indeed more than 0.10.

Conclusion

In this article, we used the formula $z=\frac{\hat{p}-p}{\sqrt{\frac{pq}{n}}}$ to calculate the test statistic $z$ for hypothesis testing of proportions. We applied this formula to a real-world example and found that the observed sample proportion is significantly higher than the null hypothesis value. This result provides strong evidence to reject the null hypothesis and conclude that the proportion of accidental deaths of the elderly attributable to residential falls is indeed more than 0.10.

References

• [1] Moore, D. S., & McCabe, G. P. (2017). Introduction to the practice of statistics. W.H. Freeman and Company.
• [2] Larson, R. J., & Farber, B. A. (2017). Elementary statistics: Picturing the world. Cengage Learning.

Additional Resources

• [1] Khan Academy. (n.d.). Hypothesis testing. Retrieved from https://www.khanacademy.org/math/statistics-probability/statistical-inference/hypothesis-testing/v/hypothesis-testing
• [2] Stat Trek. (n.d.). Hypothesis testing. Retrieved from https://stattrek.com/hypothesis-test/hypothesis-testing.aspx
  Frequently Asked Questions (FAQs) about Hypothesis Testing for Proportions ====================================================================

Q: What is hypothesis testing for proportions?

A: Hypothesis testing for proportions is a statistical method used to determine whether there is enough evidence to reject a null hypothesis about a population proportion. It involves calculating a test statistic, such as the z-score, and comparing it to a critical value or a p-value.

Q: What is the null hypothesis in hypothesis testing for proportions?

A: The null hypothesis in hypothesis testing for proportions is typically a statement about the population proportion, such as "p = 0.10" or "p ≤ 0.10". The alternative hypothesis is a statement about the population proportion that is not equal to the null hypothesis.

Q: How do I calculate the sample proportion?

A: To calculate the sample proportion, you need to know the number of successful outcomes (i.e., the number of individuals who exhibit the characteristic of interest) and the total sample size. The sample proportion is calculated by dividing the number of successful outcomes by the total sample size.

Q: What is the formula for calculating the test statistic z?

A: The formula for calculating the test statistic z is:

z = (p̂ - p) / √(pq/n)

where p̂ is the sample proportion, p is the population proportion, q is the population proportion of the complement (i.e., q = 1 - p), and n is the sample size.

Q: How do I interpret the results of a hypothesis test for proportions?

A: To interpret the results of a hypothesis test for proportions, you need to consider the p-value and the test statistic z. If the p-value is less than a certain significance level (e.g., 0.05), you can reject the null hypothesis and conclude that the population proportion is not equal to the null hypothesis value. If the p-value is greater than the significance level, you cannot reject the null hypothesis.

Q: What is the difference between a one-tailed and a two-tailed test?

A: A one-tailed test is used when the alternative hypothesis is directional (i.e., the population proportion is either greater than or less than the null hypothesis value). A two-tailed test is used when the alternative hypothesis is non-directional (i.e., the population proportion is not equal to the null hypothesis value).

Q: How do I choose the significance level for a hypothesis test?

A: The significance level is a threshold for determining whether to reject the null hypothesis. Common significance levels include 0.05, 0.01, and 0.10. The choice of significance level depends on the research question and the level of precision desired.

Q: What is the relationship between the sample size and the power of a hypothesis test?

A: The sample size affects the power of a hypothesis test. A larger sample size increases the power of the test, making it more likely to detect a statistically significant effect.

Q: How do I calculate the power of a hypothesis test?

A: The power of a hypothesis test can be calculated using a power curve or a power table. The power curve shows the probability of detecting a statistically significant effect as a function of the sample size and the effect size.

Conclusion

Hypothesis testing for proportions is a powerful statistical method for determining whether there is enough evidence to reject a null hypothesis about a population proportion. By understanding the basics of hypothesis testing for proportions, you can apply this method to a wide range of research questions and make informed decisions about your data.