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Significance Tests / Hypothesis Testing

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.

Hypothesis Testing on the TI – 83

But if your aim is to make Type I errors (rejecting the null hypothesis when it is true) less than a certain proportion of times then you need something like an $lpha$, and given that approach if you want to minimise Type II errors (failing to reject the null hypothesis when it is false) then you need to reject when you have extreme values of the test statistic as shown by the $p$-value which are suggestive of the alternative hypothesis.

Five Steps in a Hypothesis Test

Null and Alternative Hypotheses for a Mean

If the absolute value of the t-value is greater than the critical value, you reject the null hypothesis. If the absolute value of the t-value is less than the critical value, you fail to reject the null hypothesis. You can calculate the critical value in Minitab or find the critical value from a t-distribution table in most statistics books. For more information calculating the critical value in Minitab, go to and click Use the ICDF to calculate critical values.

A test statistic is a standardized value that is calculated from sample data during a hypothesis test. You can use test statistics to determine whether to reject the null hypothesis. The test statistic compares your data with what is expected under the null hypothesis. The test statistic is used to calculate the p-value.

Here are the five steps of the test of hypothesis:

The test statistic for examining hypotheses about one population mean:

NOTE: Excel can actually find the value of the CHI-SQUARE. To find this value first select an empty cell on the spread sheet then in the formula bar type "=CHIINV(D12,2)." D12 designates the p-Value found previously and 2 is the degrees of freedom (number of rows minus one). The CHI-SQUARE value in this case is 12.07121. If we refer to the CHI-SQUARE table we will see that the cut off is 4.60517 since 12.07121>4.60517 we reject the null. The following screen shot shows you how to the CHI-SQUARE value.

When you have found the F value, you can compare it with an f critical value in the table. If your observed value of F is larger than the value in the F table, then you can reject the with 95 percent confidence that the between your two populations isn’t due to random chance.

Let's return finally to the question of whether we reject or fail to reject the null hypothesis.
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Since the p-value ll hypothesis with an alpha value of 0.05.

where the observed sample mean, μ0 = value specified in null hypothesis, s = standard deviation of the sample measurements and n = the number of differences.

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

Note: P( z > 2.629) = .0043, this area/probabilityshould be doubled to get the correct p-value, the calculator has already doubledthe area for you, 2(.0043) = .0086 Since p = .008 is less than .05 wewould reject the Null hypothesis at the .05 level.

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

Imagine an enclosure in a zoo where you can't see its inhabitants. You want to test the hypothesis that it is inhabited by monkeys by putting a banana into the cage and check if it is gone the next day. This is repeated N times for enhanced statistical significance.

Next section: to Inferential statistics (testing hypotheses)

Note that if the alternative hypothesis is the less-than alternative, you reject H0 only if the test statistic falls in the left tail of the distribution (below –2). Similarly, if Ha is the greater-than alternative, you reject H0 only if the test statistic falls in the right tail (above 2).

“Accept null hypothesis” or “fail ..

Now you can formulate a null hypothesis: Given that there are monkeys in the enclosure, it is very probable that they will find and eat the banana, so if the bananas are untouched each day, it is very improbable that there are any monkeys inside.

fail to reject the null hypothesis or that we don't have ..

where the observed sample mean difference, μ0 = value specified in null hypothesis, sd = standard deviation of the differences in the sample measurements and n = sample size. For instance, if we wanted to test for a difference in mean SAT Math and mean SAT Verbal scores, we would random sample subjects, record their SATM and SATV scores in two separate columns, then create a third column that contained the differences between these scores. Then the sample mean and sample standard deviation would be those that were calculated on this column of differences.

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