Given That A Test Statistic In A Left-tailed Test Is Z = − 1.25 Z = -1.25 Z = − 1.25 , Use A 0.05 Significance Level To Find The P P P -value And State The Conclusion About The Null Hypothesis.A. 0.1056; Reject The Null Hypothesis B. 0.1056; Fail To Reject

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Introduction

In statistical hypothesis testing, the P-value is a crucial concept that helps us determine the probability of observing a test statistic as extreme or more extreme than the one we obtained, assuming that the null hypothesis is true. In this article, we will explore how to find the P-value for a left-tailed test and use it to make a conclusion about the null hypothesis.

What is a Left-Tailed Test?

A left-tailed test is a type of hypothesis test where we are interested in determining whether the population mean is less than a certain value. This type of test is used when we expect the population mean to be less than the hypothesized value. In a left-tailed test, we reject the null hypothesis if the test statistic is less than a certain critical value.

Given Information

We are given a test statistic in a left-tailed test, which is z=−1.25z = -1.25. We are also given a significance level of 0.05. Our goal is to find the P-value and use it to make a conclusion about the null hypothesis.

Finding the P-Value

To find the P-value, we need to use a standard normal distribution (Z-distribution) table or calculator. The P-value is the probability of observing a test statistic as extreme or more extreme than the one we obtained, assuming that the null hypothesis is true.

Using a standard normal distribution table or calculator, we find that the P-value for a left-tailed test with a test statistic of z=−1.25z = -1.25 is approximately 0.1056.

Interpreting the P-Value

The P-value of 0.1056 represents the probability of observing a test statistic as extreme or more extreme than z=−1.25z = -1.25, assuming that the null hypothesis is true. In this case, the P-value is greater than the significance level of 0.05.

Conclusion

Since the P-value (0.1056) is greater than the significance level (0.05), we fail to reject the null hypothesis. This means that we do not have enough evidence to conclude that the population mean is less than the hypothesized value.

Why Fail to Reject the Null Hypothesis?

Failing to reject the null hypothesis does not necessarily mean that the null hypothesis is true. It simply means that we do not have enough evidence to reject it. In this case, the P-value is greater than the significance level, which means that the observed test statistic is not statistically significant.

Implications of Failing to Reject the Null Hypothesis

Failing to reject the null hypothesis has several implications. It means that we do not have enough evidence to conclude that the population mean is less than the hypothesized value. This can be due to several reasons, such as:

  • The sample size is too small to detect a significant effect.
  • The data is not reliable or accurate.
  • The null hypothesis is true, and there is no effect.

Conclusion

In conclusion, we have found the P-value for a left-tailed test with a test statistic of z=−1.25z = -1.25 and a significance level of0.05. The P-value is approximately 0.1056, which is greater than the significance level. Therefore, we fail to reject the null hypothesis. This means that we do not have enough evidence to conclude that the population mean is less than the hypothesized value.

References

  • Moore, D. S., & McCabe, G. P. (2012). Introduction to the practice of statistics. W.H. Freeman and Company.
  • Rosner, B. (2010). Fundamentals of biostatistics. Cengage Learning.

Additional Resources

Introduction

In our previous article, we discussed how to find the P-value and make a conclusion about the null hypothesis in a left-tailed test. In this article, we will answer some frequently asked questions about P-values and null hypotheses in left-tailed tests.

Q: What is the P-value, and how is it used in hypothesis testing?

A: The P-value is the probability of observing a test statistic as extreme or more extreme than the one we obtained, assuming that the null hypothesis is true. It is used in hypothesis testing to determine whether the observed test statistic is statistically significant.

Q: How do I find the P-value for a left-tailed test?

A: To find the P-value for a left-tailed test, you can use a standard normal distribution (Z-distribution) table or calculator. The P-value is the probability of observing a test statistic as extreme or more extreme than the one we obtained, assuming that the null hypothesis is true.

Q: What is the significance level, and how is it used in hypothesis testing?

A: The significance level is the maximum probability of rejecting the null hypothesis when it is true. It is used in hypothesis testing to determine whether the observed test statistic is statistically significant.

Q: How do I determine whether to reject or fail to reject the null hypothesis?

A: To determine whether to reject or fail to reject the null hypothesis, you need to compare the P-value to the significance level. If the P-value is less than the significance level, you reject the null hypothesis. If the P-value is greater than the significance level, you fail to reject the null hypothesis.

Q: What is the difference between rejecting and failing to reject the null hypothesis?

A: Rejecting the null hypothesis means that you have enough evidence to conclude that the population mean is not equal to the hypothesized value. Failing to reject the null hypothesis means that you do not have enough evidence to conclude that the population mean is not equal to the hypothesized value.

Q: Why is it important to consider the P-value and significance level in hypothesis testing?

A: It is important to consider the P-value and significance level in hypothesis testing because they help you determine whether the observed test statistic is statistically significant. If the P-value is less than the significance level, you can be confident that the observed test statistic is statistically significant.

Q: Can I use the P-value and significance level to make conclusions about the population mean?

A: Yes, you can use the P-value and significance level to make conclusions about the population mean. If the P-value is less than the significance level, you can conclude that the population mean is not equal to the hypothesized value. If the P-value is greater than the significance level, you cannot conclude that the population mean is not equal to the hypothesized value.

Q: What are some common mistakes to avoid when using the P-value and significance level in hypothesis testing?

A: Some common mistakes to avoid when using the P-value and significance level in hypothesis testing include:

  • Failing to consider the sample size and data quality.
  • Using an incorrect significance level.
  • Failing to interpret the P-value correctly.
  • Drawing conclusions based on a single test statistic.

Conclusion

In conclusion, the P-value and significance level are crucial concepts in hypothesis testing. By understanding how to find the P-value and use it to make conclusions about the null hypothesis, you can make informed decisions about your data. Remember to consider the sample size and data quality, use an appropriate significance level, and interpret the P-value correctly.

References

  • Moore, D. S., & McCabe, G. P. (2012). Introduction to the practice of statistics. W.H. Freeman and Company.
  • Rosner, B. (2010). Fundamentals of biostatistics. Cengage Learning.

Additional Resources