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A small (p)-valueis an indication that the null hypothesis is false.

The , , is a statement of what a statistical hypothesis test is set up to establish.

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How to Determine a p-Value When Testing a Null Hypothesis

The P-value approach involves determining "likely" or "unlikely" by determining the probability — assuming the null hypothesis were true — of observing a more extreme test statistic in the direction of the alternative hypothesis than the one observed. If the P-value is small, say less than (or equal to) α, then it is "unlikely." And, if the P-value is large, say more than α, then it is "likely."

 Null and Alternative Hypotheses for a Mean

The alternative hypothesis (H1) is the opposite of the null hypothesis; in plain language terms this is usually the hypothesis you set out to investigate. For example, question is "is there a significant (not due to chance) difference in blood pressures between groups A and B if we give group A the test drug and group B a sugar pill?" and alternative hypothesis is " there is a difference in blood pressures between groups A and B if we give group A the test drug and group B a sugar pill".

The null and alternative hypotheses are:

The test statistic for examining hypotheses about one population mean:

Because the P-value 0.055 is (just barely) greater than the significance level α = 0.05, we barely fail to reject the null hypothesis. Again, we would say that there is insufficient evidence at the α = 0.05 level to conclude that the sample proportion differs significantly from 0.90.

Because this is a two-sided alternative hypothesis, the p-value is the combined area to the right of 2.47 and the left of −2.47 in a t-distribution with 35 – 1 = 34 degrees of freedom.

Null hypothesis: μ = 72 Alternative hypothesis: μ ≠72

To get the p-value on a TI 83, run a hypothesis test. Watch the video or read below.

Would you reject the hypothesis H(0):MU = 72 versus the alternative H(1):MU =/= 72 on the basis of the observations, when testing at level ALPHA = .05?

In a test of H(0): MU = 5 versus H(1): MU =/= 5 at ALPHA = .01, we would reject H(0) if: (a) XBAR - 5 > 1.29 or 5 - XBAR > 1.29 (b) XBAR - 5 > 7.74 or 5 - XBAR > 7.74 (c) XBAR - 5 > 1.29 or 5 - XBAR 1287-2

The following table shows the relationship between power and error in hypothesis testing:
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  • which is equivalent to rejecting the null hypothesis:

    The final conclusion once the test has been carried out is always given in terms of the null hypothesis.

  • which is equivalent to rejecting the null hypothesis:

    Specifically, the four steps involved in using the critical value approach to conducting any hypothesis test are:

  • which is equivalent to rejecting the null hypothesis:

    A large -value ( 0.05) indicates weak evidence against the null hypothesis, so you fail to reject the null hypothesis.

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which is equivalent to rejecting the null hypothesis:

It is a convention that a test using a t-statistic is called a t-test. That is, hypothesis tests using the above would be referred to as "1-sample t test".

the null hypothesis is rejected when it is true b.

Recall that a p-value is the probability that the test statistic would "lean" as much (or more) toward the alternative hypothesis as it does if the real truth is the null hypothesis.

the null hypothesis is not rejected when it is false c.

If we conclude "do not reject ", this does not necessarily mean that the null hypothesis is true, it only suggests that there is not sufficient evidence against in favor of ; rejecting the null hypothesis then, suggests that the alternative hypothesis may be true.

the null hypothesis is probably wrong b.

When testing hypotheses about a mean or mean difference, a t-distribution is used to find the p-value. This is a close cousin to the normal curve. T-Distributions are indexed by a quantity called degrees of freedom, calculated as df = n – 1 for the situation involving a test of one mean or test of mean difference.

the result would be unexpected if the null hypothesis were true c.

In theory we could do that, but in practice we are usually only dealing with one study and therefore only one between-group difference in cholesterol. (That’s pretty humbling when you realize that some of these trials can cost several millions of dollars). To the novice, it would seem that this one number (a difference in mean cholesterol values between two groups) might be pretty useless, but in fact it is not at all useless. We can use our knowledge of the central limit theorem to test whether the difference in average cholesterol values between the treated and placebo groups was too big (positive or negative) to be consistent with the null hypothesis of no difference in average cholesterol. This is another way of saying that

the null hypothesis is probably true d.

Notice that the top part of the statistic is the difference between the sample mean and the null hypothesis. The bottom part of the calculation is the standard error of the mean.

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