one-tailed hypothesis

Examples of one-tailed hypothesis in the following topics:

• Directional Hypotheses and One-Tailed Tests

  • A one-tailed hypothesis is one in which the value of a parameter is either above or equal to a certain value or below or equal to a certain value.
  • The vast majority of hypothesis tests involve either a point hypothesis, two-tailed hypothesis or one-tailed hypothesis.
  • A one-tailed hypothesis is a hypothesis in which the value of a parameter is specified as being either:
  • An example of a one-tailed null hypothesis, in the medical context, would be that an existing treatment, $A$, is no worse than a new treatment, $B$.
  • A one-tailed test, showing the $p$-value as the size of one tail.

• One- and Two-Tailed Tests

  • A probability calculated in only one tail of the distribution is called a "one-tailed probability. "
  • The null hypothesis for the two-tailed test is π = 0.5.
  • By contrast, the null hypothesis for the one-tailed test is π ≤ 0.5.
  • The one-tailed hypothesis is rejected only if the sample proportion is much greater than 0.5.
  • The alternative hypothesis in the two-tailed test is π ≠ 0.5.

• Steps in Hypothesis Testing

  • Be able to state the null hypothesis for both one-tailed and two-tailed tests
  • The first step is to specify the null hypothesis.
  • For a two-tailed test, the null hypothesis is typically that a parameter equals zero although there are exceptions.
  • For a one-tailed test, the null hypothesis is either that a parameter is greater than or equal to zero or that a parameter is less than or equal to zero.
  • Failure to reject the null hypothesis does not constitute support for the null hypothesis.

• Interpreting Significant Results

  • Consider the one-tailed test in the James Bond case study: Mr.
  • The null hypothesis for this one-tailed test is that π ≤ 0.5 where π is the probability of being correct on any given trial.
  • Bond is better than chance on this task.
  • Now consider the two-tailed test used in the Physicians' Reactions case study.
  • The validity of concluding the direction of the effect is clear if you note that a two-tailed test at the 0.05 level is equivalent to two separate one-tailed tests each at the 0.025 level.

• The Null and the Alternative

  • In the hypothesis testing approach of Jerzy Neyman and Egon Pearson, a null hypothesis is contrasted with an alternative hypothesis, and these are decided between on the basis of data, with certain error rates.
  • One-tailed directional.
  • A one-tailed directional alternative hypothesis is concerned with the region of rejection for only one tail of the sampling distribution.
  • Two-tailed directional.
  • A two-tailed directional alternative hypothesis is concerned with both regions of rejection of the sampling distribution.

• Exercises

  • What is the null hypothesis?
  • The probability value is computed assuming that the null hypothesis is true.
  • A study comparing a drug with a placebo on the amount of pain relief.
  • (A one-tailed test was used. )
  • Would a sample value of M= 60 be significant in a two-tailed test at the .05 level?

• Creating a Hypothesis Test

  • Decide between a one-tailed or a two-tailed statistical test.
  • A one-tailed test assesses whether the observed results are either significantly higher or smaller than the null hypothesis, but not both.
  • A two-tailed test, on the other hand, assesses both possibilities at once.
  • It achieves so by dividing the total level of significance between both tails, which also implies that it is more difficult to get significant results than with a one-tailed test.
  • This image shows a graph representation of a two-tailed hypothesis test.

• Factors Affecting Power

  • Power is higher with a one-tailed test than with a two-tailed test as long as the hypothesized direction is correct.
  • A one-tailed test at the 0.05 level has the same power as a two-tailed test at the 0.10 level.
  • A one-tailed test, in effect, raises the significance level.
  • The relationship between μ and power for H0: μ = 75, one-tailed α = 0.05, for σ's of 10 and 15.
  • The relationship between significance level and power with one-tailed tests: μ = 75, real μ = 80, and σ = 10

• Two-sided hypothesis testing with p-values

  • In one-sided tests, we shade the single tail in the direction of the alternative hypothesis.
  • p-value = left tail + right tail = 2 × (left tail) = 0.4180
  • If the null hypothesis is true, we incorrectly reject the null hypothesis about 5% of the time when the sample mean is above the null value, as shown in Figure 4.19.
  • Then if we change to a one-sided test, we would use H A : µ < µ 0 .
  • If the null hypothesis is true, then we would observe such a case about 5% of the time.

• Testing a Single Proportion

  • Do you have one group or two groups?
  • Therefore, we can perform a one proportion $z$-test to test this belief.
  • There is no reason to think how many cell phones one household owns has any bearing on the next household.
  • Determine the Critical Region(s): Based on our hypotheses are we performing a left-tailed, right tailed or two-tailed test?
  • We will perform a right-tailed test, since we are only concerned with the proportion being more than 30% of households.