Using Confidence Intervals to Test Hypotheses When analyzing the last digits of telephone numbers in Port Jefferson, it is found that among 1000 randomly selected digits, 119 are zeros. If the digits are randomly selected, the proportion of zeros should be 0.1.

a. Use the critical value method with a 0.05 significance level to test the claim that the proportion of zeros equals 0.1.

검증된 단계별 안내

Step 1: Define the null hypothesis (H₀) and the alternative hypothesis (Hₐ). The null hypothesis is H₀: p = 0.1, which states that the proportion of zeros is 0.1. The alternative hypothesis is Hₐ: p ≠ 0.1, which states that the proportion of zeros is not 0.1. This is a two-tailed test.

Step 2: Calculate the sample proportion (p̂). The sample proportion is given by p̂ = x / n, where x is the number of zeros observed (119) and n is the total number of digits sampled (1000). Substitute the values to find p̂.

Step 3: Compute the standard error (SE) of the sample proportion. The formula for the standard error is SE = sqrt((p₀ * (1 - p₀)) / n), where p₀ is the hypothesized population proportion (0.1) and n is the sample size (1000). Substitute the values to calculate SE.

Step 4: Calculate the test statistic (z). The formula for the z-test statistic is z = (p̂ - p₀) / SE, where p̂ is the sample proportion, p₀ is the hypothesized proportion, and SE is the standard error. Substitute the values to compute z.

Step 5: Determine the critical z-values for a two-tailed test at a 0.05 significance level. The critical z-values are ±1.96. Compare the calculated z-value to the critical z-values. If the calculated z-value falls outside the range of -1.96 to 1.96, reject the null hypothesis. Otherwise, fail to reject the null hypothesis.

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Confidence Intervals

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the true population parameter. It provides an estimate of uncertainty around a sample statistic, allowing researchers to make inferences about the population. For hypothesis testing, confidence intervals can help determine if a sample proportion significantly differs from a hypothesized value.

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about population parameters based on sample data. It involves formulating a null hypothesis (H0) and an alternative hypothesis (H1), then using sample data to determine whether to reject H0. The significance level, often set at 0.05, indicates the probability of rejecting H0 when it is true, guiding the decision-making process.

Critical Value Method

The critical value method is a technique used in hypothesis testing to determine the threshold at which the null hypothesis can be rejected. It involves calculating a test statistic from sample data and comparing it to a critical value derived from a statistical distribution (e.g., z-distribution for proportions). If the test statistic exceeds the critical value, the null hypothesis is rejected, indicating a significant difference.