Suppose That You Want To Perform A Test Of H 0 : P = 0.7 H_0: P=0.7 H 0 ​ : P = 0.7 Versus H A : P ≠ 0.7 H_a: P \neq 0.7 H A ​ : P  = 0.7 . An SRS Of Size 80 From The Population Yields 59 Successes. Calculate The Standardized Test Statistic And P P P -value. Assume That The Conditions

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Introduction

In statistics, hypothesis testing is a method of making inferences about a population parameter based on a sample of data. The goal of hypothesis testing is to determine whether there is sufficient evidence to reject a null hypothesis, which is a statement of no effect or no difference. In this article, we will discuss how to perform a test of $H_0: p=0.7$ versus $H_a: p \neq 0.7$ using a standardized test statistic and $P$-value.

Null and Alternative Hypotheses

The null hypothesis, $H_0: p=0.7$, states that the true population proportion is equal to 0.7. The alternative hypothesis, $H_a: p \neq 0.7$, states that the true population proportion is not equal to 0.7. This is a two-tailed test, meaning that we are interested in detecting any difference between the true population proportion and 0.7, regardless of whether it is an increase or a decrease.

Sample Data

We are given a simple random sample (SRS) of size 80 from the population, which yields 59 successes. This means that 59 out of 80 individuals in the sample have the characteristic of interest.

Standardized Test Statistic

The standardized test statistic, also known as the z-statistic, is a measure of how many standard deviations away from the null hypothesis the sample data are. It is calculated using the following formula:

$$z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}$$

where $\hat{p}$ is the sample proportion, $p_0$ is the null hypothesis proportion, and $n$ is the sample size.

In this case, we have:

$$\hat{p} = \frac{59}{80} = 0.7375$$

$$p_0 = 0.7$$

$$n = 80$$

Plugging these values into the formula, we get:

$$z = \frac{0.7375 - 0.7}{\sqrt{\frac{0.7(1-0.7)}{80}}}$$

$$z = \frac{0.0375}{\sqrt{\frac{0.21}{80}}}$$

$$z = \frac{0.0375}{0.0313}$$

$$z = 1.2$$

P-Value

The $P$-value is the probability of observing a test statistic at least as extreme as the one we obtained, assuming that the null hypothesis is true. In this case, we are interested in the probability of observing a z-statistic of at least 1.2 or at most -1.2, assuming that the null hypothesis is true.

Using a standard normal distribution table or calculator, we find that the $P$-value is:

$$P = 2 \times P(Z \geq 1.2)$$

$$P = 2 \times 0.1151$$

$$P = 0.2302$$

Interpretation

The standardized test statistic, $z = 1.2$, indicates that the sample data are 1.2 standard deviations away from the null hypothesis. The $P$-value, $P = 0.2302$, indicates that there is a 23.02% chance of observing a test statistic at least as extreme as the one we obtained, assuming that the null hypothesis is true.

Since the $P$-value is greater than 0.05, we fail to reject the null hypothesis. This means that we do not have sufficient evidence to conclude that the true population proportion is not equal to 0.7.

Conclusion

In this article, we discussed how to perform a test of $H_0: p=0.7$ versus $H_a: p \neq 0.7$ using a standardized test statistic and $P$-value. We calculated the standardized test statistic, $z = 1.2$, and the $P$-value, $P = 0.2302$. Since the $P$-value is greater than 0.05, we fail to reject the null hypothesis. This means that we do not have sufficient evidence to conclude that the true population proportion is not equal to 0.7.

References

• Moore, D. S., & McCabe, G. P. (2013). Introduction to the practice of statistics. W.H. Freeman and Company.
• Rosner, B. (2010). Fundamentals of biostatistics. Cengage Learning.

Glossary

• Null hypothesis: A statement of no effect or no difference.
• Alternative hypothesis: A statement of an effect or a difference.
• Standardized test statistic: A measure of how many standard deviations away from the null hypothesis the sample data are.
• P-value: The probability of observing a test statistic at least as extreme as the one we obtained, assuming that the null hypothesis is true.
  

Q: What is hypothesis testing?

A: Hypothesis testing is a method of making inferences about a population parameter based on a sample of data. The goal of hypothesis testing is to determine whether there is sufficient evidence to reject a null hypothesis, which is a statement of no effect or no difference.

Q: What are the types of hypothesis testing?

A: There are two main types of hypothesis testing:

• One-tailed test: A one-tailed test is used to determine whether there is a difference between the true population parameter and a specified value in one direction (e.g., whether the true population mean is greater than a specified value).
• Two-tailed test: A two-tailed test is used to determine whether there is a difference between the true population parameter and a specified value in both directions (e.g., whether the true population mean is greater than or less than a specified value).

Q: What is the null hypothesis?

A: The null hypothesis is a statement of no effect or no difference. It is a default assumption that there is no effect or no difference between the true population parameter and a specified value.

Q: What is the alternative hypothesis?

A: The alternative hypothesis is a statement of an effect or a difference. It is a statement that there is an effect or a difference between the true population parameter and a specified value.

Q: What is the p-value?

A: The p-value is the probability of observing a test statistic at least as extreme as the one we obtained, assuming that the null hypothesis is true. It is a measure of the strength of the evidence against the null hypothesis.

Q: How do I interpret the p-value?

A: The p-value can be interpreted as follows:

• p-value < 0.05: The null hypothesis is rejected, and there is sufficient evidence to conclude that the true population parameter is not equal to the specified value.
• 0.05 < p-value < 0.10: The null hypothesis is not rejected, but there is some evidence to suggest that the true population parameter may not be equal to the specified value.
• p-value > 0.10: The null hypothesis is not rejected, and there is not enough evidence to conclude that the true population parameter is not equal to the specified value.

Q: What is the standardized test statistic?

A: The standardized test statistic, also known as the z-statistic, is a measure of how many standard deviations away from the null hypothesis the sample data are. It is calculated using the following formula:

$$z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}$$

where $\hat{p}$ is the sample proportion, $p_0$ is the null hypothesis proportion, and $n$ is the sample size.

Q: How do I calculate the standardized test statistic?

A: To calculate the standardized test statistic, you need to know the sample proportion, the null hypothesis proportion, and the sample size. You can use the following formula:

$$z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}$$

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

A: A one-tailed test is used to determine whether there is a difference between the true population parameter and a specified value in one direction (e.g., whether the true population mean is greater than a specified value). A two-tailed test is used to determine whether there is a difference between the true population parameter and a specified value in both directions (e.g., whether the true population mean is greater than or less than a specified value).

Q: How do I choose between a one-tailed and a two-tailed test?

A: You should choose between a one-tailed and a two-tailed test based on the research question and the type of data you are working with. If you are interested in determining whether there is a difference between the true population parameter and a specified value in one direction, you should use a one-tailed test. If you are interested in determining whether there is a difference between the true population parameter and a specified value in both directions, you should use a two-tailed test.

Q: What is the significance level?

A: The significance level, also known as the alpha level, is the maximum probability of rejecting the null hypothesis when it is true. It is usually set at 0.05, but it can be set at any value between 0 and 1.

Q: How do I choose the significance level?

A: You should choose the significance level based on the research question and the type of data you are working with. A smaller significance level (e.g., 0.01) is more conservative and will result in fewer Type I errors, but it may also result in fewer Type II errors. A larger significance level (e.g., 0.10) is less conservative and will result in more Type I errors, but it may also result in more Type II errors.

Q: What is the power of a test?

A: The power of a test is the probability of rejecting the null hypothesis when it is false. It is a measure of the ability of the test to detect a true effect.

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

A: To calculate the power of a test, you need to know the significance level, the sample size, and the effect size. You can use the following formula:

$$\text{Power} = 1 - \beta$$

where $\beta$ is the probability of a Type II error.

Q: What is the effect size?

A: The effect size is a measure of the magnitude of the effect being tested. It is usually expressed as a correlation coefficient (e.g., r) or a standardized mean difference (e.g., d).

Q: How do I choose the effect size?

A: You should choose the effect size based on the research question and the type of data you are working with. A larger effect size (e.g., r = 0.5) will result in a more powerful test, but it may also result in a more conservative test.

Q: What is the relationship between the p-value and the effect size?

A: The p-value and the effect size are related in that a smaller p-value (e.g., p < 0.05) indicates a larger effect size (e.g., r = 0.5). However, the relationship between the p-value and the effect size is not always straightforward, and it can be influenced by various factors, such as the sample size and the significance level.

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

A: To interpret the results of a hypothesis test, you need to consider the p-value, the effect size, and the sample size. If the p-value is less than the significance level (e.g., p < 0.05), you can conclude that the null hypothesis is rejected, and there is sufficient evidence to support the alternative hypothesis. If the p-value is greater than the significance level (e.g., p > 0.05), you cannot conclude that the null hypothesis is rejected, and there is not enough evidence to support the alternative hypothesis.

Q: What are the limitations of hypothesis testing?

A: Hypothesis testing has several limitations, including:

• Assumes normality: Hypothesis testing assumes that the data are normally distributed, which may not always be the case.
• Assumes independence: Hypothesis testing assumes that the data are independent, which may not always be the case.
• Assumes equal variances: Hypothesis testing assumes that the variances of the groups are equal, which may not always be the case.
• May not detect small effects: Hypothesis testing may not detect small effects, which can be a problem in many research areas.

Q: What are the alternatives to hypothesis testing?

A: There are several alternatives to hypothesis testing, including:

• Confidence intervals: Confidence intervals provide a range of values within which the true population parameter is likely to lie.
• Regression analysis: Regression analysis is a method of modeling the relationship between a dependent variable and one or more independent variables.
• Non-parametric tests: Non-parametric tests are used when the data do not meet the assumptions of parametric tests.

Q: What is the difference between a parametric and a non-parametric test?

A: A parametric test is a test that assumes that the data are normally distributed and that the variances of the groups are equal. A non-parametric test is a test that does not assume that the data are normally distributed and that the variances of the groups are equal.

Q: How do I choose between a parametric and a non-parametric test?

A: You should choose between a parametric and a non-parametric test based on the research question and the type of data you are working with. If the data meet the assumptions of parametric tests (e.g., normality, equal variances), you should use a parametric test. If the data do not meet the assumptions of parametric tests, you should use a non-parametric test.

Q: What is the difference between a paired and an independent samples test?

A: A paired samples test is a test that compares the means of two related groups (e.g., before and after treatment). An independent samples test is a test that compares the means of two unrelated groups (e