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Left-tailed tests are statistical tests that assess if a sample mean is significantly less than a population mean. Right-tailed tests assess if a sample mean is significantly greater than a population mean. Left-tailed tests have the rejection region on the left side of the distribution, while right-tailed tests have the rejection region on the right side. In other words, left-tailed tests reject the null hypothesis when the sample mean is too low, while right-tailed tests reject the null hypothesis when the sample mean is too high.
In statistics, we use to determine whether some claim about a is true or not.
Whenever we perform a hypothesis test, we always write a null hypothesis and an alternative hypothesis, which take the following forms:
H0 (Null Hypothesis): Population parameter = ≤, ≥ some value
HA (Alternative Hypothesis): Population parameter <, >, ≠ some value
There are three different types of hypothesis tests:
- Two-tailed test: The alternative hypothesis contains the “≠” sign
- Left-tailed test: The alternative hypothesis contains the “<” sign
- Right-tailed test: The alternative hypothesis contains the “>” sign
Notice that we only have to look at the sign in the alternative hypothesis to determine the type of hypothesis test.
Left-tailed test: The alternative hypothesis contains the “<” sign
Right-tailed test: The alternative hypothesis contains the “>” sign
The following examples show how to identify left-tailed and right-tailed tests in practice.
Example: Left-Tailed Test
Suppose it’s assumed that the average weight of a certain widget produced at a factory is 20 grams. However, one inspector believes the true average weight is less than 20 grams.
To test this, he weighs a of 20 widgets and obtains the following information:
- n = 20 widgets
- x = 19.8 grams
- s = 3.1 grams
He then performs a hypothesis test using the following null and alternative hypotheses:
H0 (Null Hypothesis): μ ≥ 20 grams
HA (Alternative Hypothesis): μ < 20 grams
The test statistic is calculated as:
- t = (x – µ) / (s/√n)
- t = (19.8-20) / (3.1/√20)
- t = -.2885
According to the , the t critical value at α = .05 and n-1 = 19 degrees of freedom is –1.729.
Since the test statistic is not less than this value, the inspector fails to reject the null hypothesis. He does not have sufficient evidence to say that the true mean weight of widgets produced at this factory is less than 20 grams.
Example: Right-Tailed Test
Suppose it’s assumed that the average height of a certain species of plant is 10 inches tall. However, one botanist claims the true average height is greater than 10 inches.
To test this claim, she goes out and measures the height of a of 15 plants and obtains the following information:
- n = 15 plants
- x = 11.4 inches
- s = 2.5 inches
She then performs a hypothesis test using the following null and alternative hypotheses:
H0 (Null Hypothesis): μ ≤ 10 inches
HA (Alternative Hypothesis): μ > 10 inches
The test statistic is calculated as:
- t = (x – µ) / (s/√n)
- t = (11.4-10) / (2.5/√15)
- t = 2.1689
According to the , the t critical value at α = .05 and n-1 = 14 degrees of freedom is 1.761.
Since the test statistic is greater than this value, the botanist can reject the null hypothesis. She has sufficient evidence to say that the true mean height for this species of plant is greater than 10 inches.