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.


Online Friends A Pew Research Center poll of 1060 teens aged 13 to 17 showed that 57% of them have made new friends online. Use a 0.01 significance level to test the claim that half of all teens have made new friends online.

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 teens who have made new friends online is 0.5 (H₀: p = 0.5). The alternative hypothesis is that the proportion is not equal to 0.5 (H₁: p ≠ 0.5). This is a two-tailed test.
2. Step 2: Identify the sample proportion (p̂), sample size (n), and hypothesized proportion (p₀). From the problem, p̂ = 0.57, n = 1060, and p₀ = 0.5.
3. Step 3: Calculate the test statistic using the formula: z = (p̂ - p₀) / √((p₀(1 - p₀)) / n). Substitute the values of p̂, p₀, and n into the formula to compute the z-score.
4. Step 4: Determine the P-value for the calculated z-score. Since this is a two-tailed test, find the area under the standard normal curve corresponding to the z-score and double it to account for both tails.
5. Step 5: Compare the P-value to the significance level (α = 0.01). If the P-value is less than or equal to 0.01, reject the null hypothesis (H₀). Otherwise, fail to reject the null hypothesis. Based on this decision, state the conclusion about the original claim that half of all teens have made new friends online.

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 indicates the presence of an effect or difference. 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., 0.01), 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 binomial outcomes, 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.