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.


Internet Use A random sample of 5005 adults in the United States includes 751 who do not use the Internet (based on three Pew Research Center polls). Use a 0.05 significance level to test the claim that the percentage of U.S. adults who do not use the Internet is now less than 48%, which was the percentage in the year 2000. If there appears to be a difference, is it dramatic?

Verified step by step guidance

1. Step 1: Define the null hypothesis (H₀) and the alternative hypothesis (H₁). The null hypothesis represents the claim that the percentage of U.S. adults who do not use the Internet is equal to 48% (H₀: p = 0.48). The alternative hypothesis represents the claim that the percentage is now less than 48% (H₁: p < 0.48).
2. Step 2: Identify the sample proportion (p̂) and the sample size (n). The sample size is n = 5005, and the number of adults who do not use the Internet is 751. Calculate the sample proportion using the formula p̂ = x / n, where x is the number of successes (in this case, adults who do not use the Internet).
3. Step 3: Compute the test statistic using the formula z = (p̂ - p₀) / √(p₀(1 - p₀) / n), where p₀ is the hypothesized population proportion (0.48), p̂ is the sample proportion, and n is the sample size. This formula calculates the z-score, which measures how many standard deviations the sample proportion is from the hypothesized proportion.
4. Step 4: Determine the P-value. Using the z-score from Step 3, find the P-value by looking up the cumulative probability in the standard normal distribution table. Since this is a left-tailed test (H₁: p < 0.48), the P-value corresponds to the area to the left of the z-score.
5. Step 5: Compare the P-value to the significance level (α = 0.05). If the P-value is less than 0.05, reject the null hypothesis (H₀). Otherwise, fail to reject the null hypothesis. Based on this decision, state the conclusion about the null hypothesis and address whether the percentage of U.S. adults who do not use the Internet is now less than 48%.

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. In this context, the null hypothesis would state that the proportion of U.S. adults who do not use the Internet is 48% or more, while the alternative hypothesis would claim it is less than 48%.

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 as extreme as, or more extreme than, the one observed, assuming the null hypothesis is true. A smaller P-value indicates stronger evidence against the null hypothesis. In this scenario, if the P-value is less than the significance level of 0.05, it would lead to rejecting the null hypothesis in favor of the alternative.

Normal Approximation to the Binomial Distribution

The normal approximation to the binomial distribution is a technique used when dealing with large sample sizes, allowing the binomial distribution to be approximated by a normal distribution. This is applicable when both np and n(1-p) are greater than 5, where n is the sample size and p is the probability of success. In this case, the sample of 5005 adults can be analyzed using the normal distribution to simplify calculations related to proportions, such as determining the test statistic and P-value.