Tests of Hypotheses
Statistics plays a very important role in the field of research. Its different tools and techniques help a researcher draw valid and reliable conclusions about the population on the basis of the sample. It helps him decide whether to accept or eject a hypothesis after evaluating the sample.
Hypothesis is a theory, claim or assertion about a particular parameter of a population. It needs to be proven true. Once proven true, it is accepted; otherwise, it is rejected.
Types of hypothesis:
1. Null hypothesis (Ho) is always one of status quo or no difference.
Example: Ho : The mean fill per box of cereal is 368 grams. (μ = 368)
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1. A pharmaceutical firm claims that the average time for a drug to take effect is 18 minutes with a standard deviation of 2 minutes. In a sample, of 36 trials, the average time was 20 minutes. Test the claim against the alternative that the average time is not equal to 18 minutes, using a 0.01 level of significance.
H0: The average time for a drug to take effect is 18 minutes. (μ= 18 mins.)
H1: The average time for a drug to take effect is not 18 minutes . (μ ≠ 18 mins.)
The two types of alternative hypotheses are:
1. Non-directional alternative or two-sided alternative - connotes inequality as indicated by ≠.
H1: The mean fill per box is not 368 grams. ( μ ≠ 368)
H1: There is difference between males and females in their
performance in Statistics. (μm ≠ μf)
2. Directional alternative or one-sided alternative – connotes inequality as indicated by comparative adjectives like lower than (<), more than (>) and others.
H1: The mean fill per box is greater than 368 grams. (μ > 368)
H1: The mean fill per box is less than 368 grams. (μ < 368)
H1: The performance in Statistics of males is higher than the
females. (μm > μf)
Regions of Rejection and Non-rejection
The sampling distribution of the test is divided into two regions: (1) region of rejection (sometimes called critical region), and (2) region of non-rejection. (See Figure 1)
If the test statistic falls into non-rejection region, the null hypothesis cannot be rejected. If the test statistic falls into rejection region, the null hypothesis is rejected.
Risks in Decision Making Using Hypothesis-Testing Methodology.
Type I Error occurs if the null hypothesis is rejected when in fact it is true and should not be rejected.
Type II Error occurs if the null hypothesis is not rejected when in fact it is false and should be rejected.
The Level of Significance
The probability of committing a Type I error, denoted by α (lowercase Greek letter alpha), is referred to as the level of significance of the statistical test. Traditionally, one controls the Type I error rate by deciding the risk level α, he she is willing to tolerate in rejecting the null hypothesis when it is true. Because the level of significance is specified before the hypothesis test is performed, the risk of committing a Type I error, α, is directly under the control of the individual performing the test. Researchers traditionally select α level of 0.05 or smaller.
The choice of selecting a particular risk level for making a Type I error is dependent on the cost of making a Type I error. Once the value for α is specified, the size of the rejection region is known because the probability of rejection under the null hypothesis. From this fact, the critical value or values that divide the rejection and nonrejection regions are determined.