Witte & Witte Chapter 11 More about Hypothesis Testing
True Sampling Distribution
Centered about the true population mean, this distribution produces the one observed mean (or z).
Rejecting the null hypothesis
H0 is rejected whenever the observed z qualifies as a rare outcome—one that could have occurred just by chance with a probability of .05 or less— on the assumption that H0 is true - This suspiciously rare outcome implies that H0 is probably false (and conversely, that H1 is probably true)
false null hypothesis
the hypothesized population mean differs considerably from the true population mean. - there is also a high probability that the observed z will qualify as a rare outcome under the hypothesized distribution and that the false H0 will be rejected
four possible outcomes of any hypothesis test
(1) Correct decision - Retain H0 when H0 is true (2) Type I error (false alarm) - Reject H0 when H0 is true (3) Type II error (miss) - Retain H0 when H0 is False (4) Correct decision - Reject H0 when H0 is False
One or Two Tails?
a hypothesis test, if there is a concern that the true population mean differs from the hypothesized population mean only in a particular direction, use the appropriate one-tailed or directional test for extra sensitivity. Otherwise, use the more customary two-tailed or nondirectional test.
standard error of the mean
a measure of the average amount by which sample means differ, just by chance, from the population mean Dividing the observed difference by the standard error to obtain a value of z locates the original observed difference along a z scale of either common outcomes (reasonably attributable to chance - retain null) or rare outcomes (not reasonably due to chance -reject null ). *Before generalizing beyond the existing data, we must always measure the effect of chance; that is, we must obtain a value for the standard error*
one-tailed test and area of "no concern."
a one-tailed test with its single rejection region must retain H0 if the observed z deviates from the hypothesized population mean in the direction of "no concern." - a one-tailed test should be adopted only when there is absolutely no concern about deviations, even very large deviations, in one direction. If there is the slightest concern about these deviations, use a two-tailed test.
level of significance
indicates how rare an observed z must be before H0 can be rejected. -To reject H0 at the .05 level of significance implies that the observed z would have occurred, just by chance, with a probability of only .05 (one chance out of twenty) or less.
Failure to reject the null hypothesis
meaning there is not significant difference between groups or treatments Saying this better than *retaining null hypothesis* - retainment would mean that the Z qualifies as a common outcome leading to the assumption that the null is true
H1: µ > 500
one-tailed test with the upper tail critical (critical z equals 1.65). This test is specially designed to detect only whether the local population mean score exceeds the national average
The two types of incorrect decisions
rejecting a true null hypothesis (a false alarm) or retaining a false null hypothesis(a miss) —can be controlled by our selection of the level of significance and of the sample size.
Two-tailed or nondirectional test
rejection regions are located in both tails of the sampling distribution - critical z scores of ±1.96
H1: µ ≤ 500
the null hypothesis should be rejected only if the mean score for the local population is less than the national average of 500 - an observed z triggers the decision to reject H0 only if z deviates too far below the national average (z = -1.65)
sampling distribution
the sample means for all possible random outcomes
true null hypothesis
there is a high probability that the observed z will qualify as a common outcome under the hypothesized sampling distribution and that the true H0 will be retained.
Hypothesized Sampling Distribution
Centered about the hypothesized population mean, this distribution is used to generate the decision rule.
Rejection of False H0 Depends on Size of Effect
If H0 really is false, the probability of a type II error, β, and the probability of a correct decision, 1 − ß, depend on the size of the effect, that is, the difference between the true and the hypothesized population means. The smaller the effect, the higher the probability of a type II error and the lower the probability of a correct decision.
type I error (or a false alarm)
Rejecting null hypothesis when it is true Having made a decision about the null hypothesis, we never know absolutely whether that decision is correct or incorrect, unless, of course, we survey the entire population. - Even if H0 is true (and, therefore, the hypothesized distribution of z about H0 also is true), there is a slight possibility that, just by chance, the one observed z actually originates from one of the shaded rejection regions of the hypothesized distribution of z, thus causing the true H0 to be rejected
One-tailed or directional test
Rejection region is located in just one tail of the sampling distribution - critical z scores of either + 1.65 or -1.65 - the decision to reject a false H0 (in favor of the research hypothesis) is more likely to occur in the one-tailed test than in the two-tailed test.
Why the Research Hypothesis Isn't Tested Directly
- The research hypothesis (not the null hypothesis) lacks the necessary precision to be tested directly. While the null hypothesis specifies a single number Because the research hypothesis is identified with the alternative hypothesis, the decision to reject the null hypothesis, should it be made, will provide strong support for the research hypothesis, while the decision to retain the null hypothesis, should it be made, will provide, at most, weak support for the null hypothesis. - Logically, a statement such as "All cows have four legs" can never be proven in spite of a steady stream of positive instances. It only takes one negative instance—one cow with three legs—to disprove the statement. By the same token, one positive instance (common outcome) doesn't prove the null hypothesis, but one negative instance (rare outcome) disproves the null hypothesis.
effect
Any difference between a true and a hypothesized population mean.
Alpha (𝜶)
The probability of a type I error, that is, the probability of rejecting a true null hypothesis. - The probability of a correct decision equals 1 − α, that is, .95.
Beta (ß)
The probability of a type II error, that is, the probability of retaining a false null hypothesis. - equals .01 - these probabilities represent areas under the true sampling distribution found by re-expressing the critical z as a deviation from the true population mean rather than from the hypothesized population mean - complements (1 − ß) of these probabilities—aid the selection of sample size
Power (1 − β)
The probability of detecting a particular effect.
H1: µ ≠ 500
This alternative hypothesis says that the null hypothesis should be rejected if the mean score for the local population differs in either direction from the national average of 500. An observed z will qualify as a rare outcome if it deviates too far either below or above the national average (z = ±1.96)
An excessively large sample size produces
an extra-sensitive hypothesis test that detects even a very small effect that, from almost any perspective, lacks importance - could cause H0 to be rejected - a wise investigator attempts to select a sample size that, because it is not excessively large, minimizes the detection of a small, unimportant effect.
An unduly small sample size will produce
an insensitive hypothesis test (with a large standard error) that will miss even a very large, important effect. - can cause H0 to be retained
type II error (or a miss)term-6
failing to reject a false null hypothesis
power curves
help select the appropriate sample size for a particular experiment.
To increase the probability of detecting a false H0
increase the sample size. - any increase in sample size causes a reduction in the standard error of the mean. - type II error is reduced - effects are noticed
