Question

If 10 children out of 108 under the age of 2 in a poor rural community in the country have 1 + episodes of diarrhea in a year, then test if this represents a significant departure from the overall rate for the country using the critical-value method.

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Normal Theory test is also known as critical value test.

The critical value approach involves determining "likely" or "unlikely" by determining whether or not the observed test statistic is more extreme than would be expected if the null hypothesis were true. That is, it entails comparing the observed test statistic to some cutoff value, called the "critical value." If the test statistic is more extreme than the critical value, then the null hypothesis is rejected in favor of the alternative hypothesis. If the test statistic is not as extreme as the critical value, then the null hypothesis is not rejected.

Specifically, the four steps involved in using the critical value approach to conduct the critical value test

1. Specify the null and alternative hypotheses.
2. Using the sample data and assuming the null hypothesis is true, calculate the value of the test statistic. To conduct the hypothesis test for the population mean μ, we use the t-statistic t∗=x¯−μs/n which follows a t-distribution with n - 1 degrees of freedom.
3. Determine the critical value by finding the value of the known distribution of the test statistic such that the probability of making a Type I error — which is denoted α (greek letter "alpha") and is called the "significance level of the test" — is small (typically 0.01, 0.05, or 0.10).
4. For 5% significance as given in the question, 95% confidence, we can find from the Z table that critical value is 1.96  
5. Compare the test statistic to the critical value. If the test statistic is more extreme in the direction of the alternative than the critical value, reject the null hypothesis in favor of the alternative hypothesis. If the test statistic is less extreme than the critical value, do not reject the null hypothesis.