More on Hypothesis Testing...
A two-tailed test indicates that the null hypothesis should be rejected when the test value is in either of the two critical regions. For a two-tailed test, the critical region must be split into two equal parts. If α = 0.01, then half of the area, or 0.005, must be to the right of the mean and half must be to the left of the mean.
Procedure for finding the Critical Values for specific α values. . .
1. Draw the figure and indicate the appropriate region
- If the test is one-tailed left, the critical region, with an area equal to α, will be on the left side of the mean.
- If the test is one-tailed right, the critical region, with an area equal to α, will be on the right of the mean.
- If the test is two-tailed, α must be divided by 2; half of the area will be on the right side of the mean, and the other half will be on the left side.
2. Subtract the area in the critical region from 0.5000, since the table gives the area for only half of the distribution (from the mean of the z-value).(Depending upon the table, may be some variation to procedure.)
3. Look up the area in the table and find the corresponding z value, which is the Critical Value. If the exact area cannot be found, use the closest value.
Hypothesis Testing Procedure:
1. State the hypotheses.
- State the null hypothesis
- State the alternative hypothesis
2. Design the study
- Select the appropriate statistical test.
- Choose a level of significance.
- Formulate a procedure for carrying out the study, like methods of selecting the sample, collecting data, and so forth.
3. Conduct the study and collect the data.
4. Evaluate the data
- Compute the test value.
- Make a decision about the null hypothesis.
5. Summarize the results.
Procedure for Solving Hypothesis-Testing Problems
- State the hypotheses.
- Find the critical value(s) from the table.
- Compute the test value.
- Make the decision to reject or not reject the null hypothesis.
- Summarize the results.
z test and t test
The following general formula is indicative of many hypotheses tests.
Test value = (observed value) minus (expected value)
Divided by: Standard error
The observed value is the statistic (such as the mean) that is computed from the sample data. The expected value is the parameter (such as the mean) that one would expect to obtain if the null hypothesis were true. The denominator is the standard error of the statistic being tested (in this case, the standard error of the mean).
Definition of the z test:
The z test is a statistical test for the mean of a population and is used when the population is normally distributed and σ is known or n is greater than or equal to 31.
The formula is: Where = sample mean
σ=population standard deviation
For a z test, the observed value is the value of the sample mean. The expected value is the value of the population mean, assuming that the null hypothesis is true. The denominator is the standard error of the mean.
In order to understand all the concepts, you should carefully follow each step in the examples shown in the text.
The z test is used when the population is normally distributed and the population standard deviation is known. But, of course, that is often not the case, is it? In that case the standard deviation from the sample can be used, but a different test must be utilized. This test is the t test.
When the size of the sample is small, the standard deviation obtained from the sample tends to underestimate the population standard deviation. If the z test were used with s instead of σ, the test value would be larger than it should be. Thus, there would be a higher probability of rejecting the null hypothesis when it was in fact true.
Characteristics of the t distribution:
- It is bell-shaped like the normal distribution.
- It is symmetrical about the mean like the normal distribution.
- The mean, median, and mode are equal to 0 and are located at the center of the distribution.
- The curve never touches the x-axis.
But. . .
- The variance is greater than 1.
- The t distribution is actually a family of curves based upon the concept of degrees of freedom (d.f.), which is related to sample size.
- As the sample size increases, the t distribution approaches the normal distribution.
The degrees of freedom are the number of values that are free to vary after a sample statistic has been computed, and they tell which specific curve to use when a distribution consists of a family of curves.
For example, if the mean of 5 values is 10, then 4 of the 5 values are free to vary. But once 4 values are selected, the fifth value must be a specific number to get a sum of 50, since 50 ÷ 5 = 10. Hence, the degrees of freedom are 5 − 1 = 4, and this value indicates which curve to use. The symbol d.f. is generally used for degrees of freedom and is one less than the samle size, n - 1.
The t test is a statistical test for the mean of a population and is used when the population is normally distributed, σ is unknown, and n < 31.
The formula d.f. = n − 1
Finally, we look at the z test for proportions. A hypothesis test involving a population proportion can be considered as a binomial experiment when there are only two outcomes and the probability of success does not change from trial to trial.
Since the normal distribution can be used as an approximation of the binomial distribution when np ≥ 5 and nq ≥ 5, the normal distribution can be used to test hypotheses for proportions using the following formula:
or where µ = np and σ =
Hypothesis Test - Part I