Testing Claims About Proportions
In Exercises 9–32, test the given claim. Identify the null hypothesis, alternative hypothesis, test statistic, P-value, or critical value(s), then state the conclusion about the null hypothesis, as well as the final conclusion that addresses the original claim. Use the P-value method unless your instructor specifies otherwise. Use the normal distribution as an approximation to the binomial distribution, as described in Part 1 of this section.


Births A random sample of 860 births in New York State included 426 boys. Use a 0.05 significance level to test the claim that 51.2% of newborn babies are boys. Do the results support the belief that 51.2% of newborn babies are boys?

Verified step by step guidance

1. Step 1: Define the null hypothesis (H₀) and the alternative hypothesis (H₁). The null hypothesis is that the proportion of boys among newborns is 51.2%, or H₀: p = 0.512. The alternative hypothesis is that the proportion is different from 51.2%, or H₁: p ≠ 0.512. This is a two-tailed test.
2. Step 2: Calculate the sample proportion (p̂). The sample proportion is given by p̂ = x / n, where x is the number of boys (426) and n is the total number of births (860). Substitute the values to find p̂.
3. Step 3: Compute the test statistic (z). Use the formula z = (p̂ - p₀) / √(p₀(1 - p₀) / n), where p₀ is the hypothesized population proportion (0.512), p̂ is the sample proportion, and n is the sample size. Substitute the values to calculate z.
4. Step 4: Determine the P-value. Using the z-value from Step 3, find the P-value by looking up the corresponding area under the standard normal distribution curve. Since this is a two-tailed test, multiply the area by 2.
5. Step 5: Compare the P-value to the significance level (α = 0.05). If the P-value is less than or equal to 0.05, reject the null hypothesis (H₀). Otherwise, fail to reject H₀. Based on this decision, state whether the results support the claim that 51.2% of newborn babies are boys.

Key Concepts

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 (H1), which represents the claim being tested. The goal is to determine whether there is enough evidence in the sample data to reject the null hypothesis in favor of the alternative.

P-value

The P-value is a measure that helps determine the strength of the evidence against the null hypothesis. It represents the probability of obtaining a test statistic at least as extreme as the one observed, assuming the null hypothesis is true. A smaller P-value indicates stronger evidence against the null hypothesis, and if the P-value is less than the significance level (e.g., 0.05), the null hypothesis is rejected.

Normal Approximation to the Binomial Distribution

The normal approximation to the binomial distribution is a technique used when dealing with binomial experiments, particularly when the sample size is large. It allows us to use the normal distribution to approximate the probabilities of the binomial distribution, making calculations easier. This approximation is valid when both np and n(1-p) are greater than 5, where n is the number of trials and p is the probability of success.