Confidence Interval and Hypothesis Testing

Hypothesis testing is a commonly used and indispensable aspect of statistics. Also referred to as confirmatory data analysis or significance testing, hypothesis testing is a statistical terminology that alludes to the procedure of selecting probability dispersion between contesting hypotheses (Davies, 2009, p.2). The observed or experimental data is systematically used to ascertain the probability that a certain hypothesis is indeed correct. This way, the results obtained become of statistical significance as it has been proven to indeed occur (Smithson, 2003, p.3). The aim is to determine the possibility that a population parameter is likely to be true.

A hypothesis test comprises two propositions that complement each other; the null (H0) and the alternative hypotheses (H1). For instance, assuming that the test is meant to ascertain if a process means varies from zero, the null hypothesis would be the mean, which equals zero and the alternative hypothesis would be the contrary; the mean is not equal to zero. As such, samples are taken from the experimental data in order to know more about the characteristics of the given population.

Normally, the hypothesis testing process entails four steps – stating the hypothesis, setting the standard for a decision, computing the test statistic, and making inferences (David, 1995). While stating the hypothesis, the null hypothesis is the beginning point. Normally, it is assumed that the observations are a product of chance. However, the null hypothesis is assumed to be true. Then the null hypothesis is put to test to ascertain its truth. The alternative hypothesis indicates that the observations are an actual representation of facts alongside a component of chance variation (Casella & Berger, 2002, p. 109).

The next step is to determine a test statistic to be used to evaluate the actuality of the null hypothesis. Then the “p-value (the probability that a test statistic at least as meaningful as the observed statistic would be seen assuming that the null hypothesis is indeed correct) is computed” (Davies, 2009, p.3). The smaller the p-value is the greater the testimony against the null hypothesis is. Finally, the p-value is compared to an acceptable significant value (alpha value). If p equals or is smaller than the acceptable value, the null hypothesis is eliminated and the alternative hypothesis is accepted as valid (Anderson, Burnham & Thompson, 2000, p. 915).

Practically, the above theory can be attested by comparing the body mass index (BMI) of men and women. Normally, it has been established that the mean weight of men dwarfs that of women. Moreover, studies done on heights of both sexes have established that akin to the mean body weight, the mean height of men is greater than that of women. As such we are challenged to analyze the BMI which is thought to be greater in men as opposed to women. To this end, we establish our null hypothesis (H0) and say that the mean weights are equal. Our alternative hypothesis (H1) contradicts H0 and states that the means are not equal. Hence;

H0; µ=µ

H1; µ≠µ

From the appendix below the value of t = 0.1769.

At 99% confidence interval and, at 39 degrees of freedom the rejection region is given as below: t0.01, 39 = 2.708.

The calculated value (0.1769) is less than the tabulated value (2.708). Hence, this falls within the rejection region. As such, we reject H0 and state that the two means are not equal thus; the mean BMI for men is greater than that of women.

References

Anderson, D. R., Burnham, K. P., & Thompson, W. L. (2000). Null hypothesis testing: problems, prevalence, and an alternative. The Journal of Wildlife Management, 64, (4), 912-923.

Casella, G. & Berger, R. (2002). Statistical inference. High Holborn, London: Thomson Learning.

David, J. (1995). Introduction to hypothesis testing. Web.

Davies, H.T.O. (2009). What are confidence intervals and p-values? Web.

Smithson, M. 2003. Confidence intervals. Thousand Oaks, CA: Sage Publications, Inc.

Triola, M. & Triola, M. (2006). Biostatistics: For the biological and health sciences. Boston: Pearson Educational, Inc.

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