Left-Tailed Test

Definition

A left-tailed test is a statistical hypothesis test in which the alternative hypothesis specifies that the parameter of interest is less than a certain value. It is used to determine if there is sufficient evidence to conclude that the true value of the parameter is on the left side of the sampling distribution.

5 Must Know Facts For Your Next Test

1. In a left-tailed test, the null hypothesis (H0) specifies that the parameter is greater than or equal to a certain value, while the alternative hypothesis (Ha) states that the parameter is less than that value.
2. The test statistic in a left-tailed test falls in the left tail of the sampling distribution under the null hypothesis, and the p-value represents the probability of observing a test statistic this extreme or more extreme, given that the null hypothesis is true.
3. Left-tailed tests are commonly used when the researcher is interested in determining if a parameter is less than a specified value, such as when testing for a decrease in mean or proportion compared to a hypothesized value.
4. The critical region for a left-tailed test is the set of values of the test statistic that are less than the critical value, which is determined by the significance level and the sampling distribution.
5. The decision rule for a left-tailed test is to reject the null hypothesis if the test statistic is less than the critical value, indicating there is sufficient evidence to conclude the parameter is less than the hypothesized value.

Review Questions

• Explain the purpose and characteristics of a left-tailed test in the context of hypothesis testing.
  • The purpose of a left-tailed test is to determine if there is sufficient evidence to conclude that the true value of a parameter is less than a hypothesized value. In a left-tailed test, the null hypothesis (H0) specifies that the parameter is greater than or equal to a certain value, while the alternative hypothesis (Ha) states that the parameter is less than that value. The test statistic falls in the left tail of the sampling distribution under the null hypothesis, and the p-value represents the probability of observing a test statistic this extreme or more extreme, given that the null hypothesis is true. The critical region for a left-tailed test is the set of values of the test statistic that are less than the critical value, and the decision rule is to reject the null hypothesis if the test statistic is less than the critical value.
• Describe how a left-tailed test would be used in the context of the topics 9.5 Additional Information and Full Hypothesis Test Examples, 10.2 Two Population Means with Known Standard Deviations, and 10.3 Comparing Two Independent Population Proportions.
  • In the context of 9.5 Additional Information and Full Hypothesis Test Examples, a left-tailed test could be used to determine if the mean or proportion of a population is less than a hypothesized value. For example, a researcher might want to test if the mean weight of a certain product is less than the manufacturer's claimed weight. In the context of 10.2 Two Population Means with Known Standard Deviations, a left-tailed test could be used to determine if the mean of one population is less than the mean of another population with known standard deviations. This could be useful in comparing the performance of two treatments or interventions. In the context of 10.3 Comparing Two Independent Population Proportions, a left-tailed test could be used to determine if the proportion of one population is less than the proportion of another population. This could be applied to compare the success rates of two different procedures or programs.
• Analyze the role of the p-value in interpreting the results of a left-tailed test and making a decision about the null hypothesis.
  • The p-value in a left-tailed test represents the probability of observing a test statistic this extreme or more extreme, given that the null hypothesis is true. If the p-value is less than the chosen significance level, it indicates that the observed data is unlikely to have occurred if the null hypothesis were true. In this case, the researcher would reject the null hypothesis and conclude that there is sufficient evidence to support the alternative hypothesis, which states that the parameter of interest is less than the hypothesized value. Conversely, if the p-value is greater than the significance level, the researcher would fail to reject the null hypothesis, indicating that there is not enough evidence to conclude that the parameter is less than the hypothesized value. The p-value, in conjunction with the chosen significance level, is a critical factor in the decision-making process for a left-tailed test.

Related terms

Hypothesis Testing:

The process of using statistical analysis to determine whether a hypothesis about a population parameter is likely to be true or false.

Alternative Hypothesis:

The statement that is accepted if the null hypothesis is rejected, indicating there is sufficient evidence to conclude the parameter of interest is different from the hypothesized value.

Sampling Distribution:

The probability distribution of a statistic computed from a random sample, which can be used to determine the likelihood of observing a particular sample statistic.