In Exercises 1–5, (a) identify the claim and state H0 and Ha, (b) decide which nonparametric test to use, (c) find the critical value(s), (d) find the test statistic, (e) decide whether to reject or fail to reject the null hypothesis, and (f) interpret the decision in the context of the original claim.
[APPLET] A meteorologist wants to determine whether days with rain occur randomly in April in his hometown. To do so, the meteorologist records whether it rains for each day in April. The results are shown, where R represents a day with rain and N represents a day with no rain. At , can the meteorologist conclude that days with rain are not random?
N R R N N N N R N R R N R R R
N R R R R N N N N R N R N N R

Verified step by step guidance

Step 1: Identify the claim and state the null and alternative hypotheses. The claim is that days with rain are not random. The null hypothesis (H₀) assumes that days with rain occur randomly, while the alternative hypothesis (Hₐ) assumes that days with rain do not occur randomly.

Step 2: Decide which nonparametric test to use. Since the problem involves testing the randomness of a sequence of categorical data (rain vs. no rain), the runs test for randomness is appropriate.

Step 3: Find the critical value(s). Determine the significance level (α = 0.05) and use the runs test critical value table or formula to find the critical values based on the total number of observations (n₁ = number of R's, n₂ = number of N's).

Step 4: Calculate the test statistic. Count the total number of runs (a run is a sequence of consecutive identical outcomes, such as RRR or NN). Use the formula for the expected number of runs and its standard deviation to compute the z-score: z = (observed runs - expected runs) / standard deviation of runs.

Step 5: Decide whether to reject or fail to reject the null hypothesis. Compare the test statistic to the critical values. If the test statistic falls in the rejection region, reject H₀. Otherwise, fail to reject H₀. Finally, interpret the decision in the context of the original claim: if H₀ is rejected, conclude that days with rain are not random; otherwise, conclude that there is insufficient evidence to support the claim.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above

Video duration:

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about a population based on sample data. It involves formulating two competing hypotheses: the null hypothesis (H0), which represents a statement of no effect or no difference, and the alternative hypothesis (Ha), which indicates the presence of an effect or difference. The goal is to determine whether there is enough evidence to reject H0 in favor of Ha.

Nonparametric Tests

Nonparametric tests are statistical tests that do not assume a specific distribution for the data. They are particularly useful when the data do not meet the assumptions required for parametric tests, such as normality. In the context of the given question, a nonparametric test like the Chi-square test for independence could be used to analyze whether the occurrence of rainy days is random.

Critical Value and Test Statistic

The critical value is a threshold that determines the boundary for rejecting the null hypothesis in hypothesis testing. It is derived from the significance level (alpha) and the distribution of the test statistic. The test statistic, calculated from the sample data, is then compared to the critical value to decide whether to reject or fail to reject H0. This comparison helps in interpreting the results in the context of the original claim.

Watch next

Master Step 1: Write Hypotheses with a bite sized video explanation from Patrick

Start learning