Claim of “At Least” or “At Most”
How do the following results change?


a. Chapter Problem claim is changed to this: “At least 50% of Internet users utilize two-factor authentication to protect their online data.”

Verified step by step guidance

1. Identify the type of claim being made. The phrase 'at least 50%' indicates a one-tailed hypothesis test where the null hypothesis (H₀) will state that the proportion of Internet users utilizing two-factor authentication is less than or equal to 50%, and the alternative hypothesis (H₁) will state that the proportion is greater than 50%.
2. Define the null and alternative hypotheses mathematically. Using p to represent the proportion of Internet users utilizing two-factor authentication: H₀: p ≤ 0.50 and H₁: p > 0.50.
3. Determine the appropriate statistical test to use. Since this is a hypothesis test for a population proportion, a z-test for proportions is typically used if the sample size is large enough to satisfy the conditions for normal approximation (np ≥ 5 and n(1-p) ≥ 5).
4. Calculate the test statistic using the formula: z = (p̂ - p₀) / √((p₀(1-p₀))/n), where p̂ is the sample proportion, p₀ is the hypothesized proportion (0.50 in this case), and n is the sample size. Ensure all values are substituted correctly.
5. Compare the calculated z-value to the critical z-value for the chosen significance level (e.g., α = 0.05). Alternatively, calculate the p-value and compare it to the significance level. If the test statistic falls in the rejection region or the p-value is less than α, reject the null hypothesis; otherwise, fail to reject it.

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to make inferences or draw conclusions about a population based on sample data. In the context of claims like 'At least 50%', it involves formulating a null hypothesis (e.g., less than 50% use two-factor authentication) and an alternative hypothesis (e.g., at least 50% use it), then using sample data to determine if there is enough evidence to reject the null hypothesis.

Confidence Intervals

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the true population parameter. When assessing claims such as 'At least 50%', confidence intervals help quantify the uncertainty around the estimate of the proportion of users employing two-factor authentication, providing a clearer picture of the data's reliability.

P-Value

The p-value is a statistical measure that helps determine the significance of results obtained in hypothesis testing. It indicates the probability of observing the sample data, or something more extreme, if the null hypothesis is true. A low p-value (typically less than 0.05) suggests strong evidence against the null hypothesis, supporting the claim that at least 50% of Internet users utilize two-factor authentication.