Testing the Difference Between Two Means In Exercises 15–24, (a) identify the claim and state Ho and Ha, (b) find the critical value(s) and identify the rejection region(s), (c) find the standardized test statistic z, (d) decide whether to reject or fail to reject the null hypothesis, and (e) interpret the decision in the context of the original claim. Assume the samples are random and independent, and the populations are normally distributed.
ACT Mathematics and Science Scores The mean ACT mathematics score for 60 high school students is 20.2. Assume the population standard deviation is 5.7. The mean ACT science score for 75 high school students is 20.6. Assume the population standard deviation is 5.9. At α=0.01, can you reject the claim that ACT mathematics and science scores are equal? (Source: ACT, Inc.)

Verified step by step guidance

1. Identify the claim and state the null hypothesis (H₀) and the alternative hypothesis (Hₐ): The claim is that the ACT mathematics and science scores are equal. This translates to H₀: μ₁ = μ₂ (the means are equal) and Hₐ: μ₁ ≠ μ₂ (the means are not equal). This is a two-tailed test.
2. Determine the critical value(s) and rejection region(s): Since this is a two-tailed test with a significance level of α = 0.01, divide α by 2 to find the area in each tail (α/2 = 0.005). Use the standard normal (Z) table to find the critical z-values corresponding to these tail areas. The rejection regions are z < -zₐ/₂ and z > zₐ/₂.
3. Calculate the standardized test statistic z: Use the formula z = (x̄₁ - x̄₂) / √((σ₁² / n₁) + (σ₂² / n₂)), where x̄₁ and x̄₂ are the sample means, σ₁ and σ₂ are the population standard deviations, and n₁ and n₂ are the sample sizes. Substitute the given values: x̄₁ = 20.2, x̄₂ = 20.6, σ₁ = 5.7, σ₂ = 5.9, n₁ = 60, and n₂ = 75.
4. Compare the calculated z-value to the critical z-values: If the calculated z-value falls in the rejection region (z < -zₐ/₂ or z > zₐ/₂), reject the null hypothesis. Otherwise, fail to reject the null hypothesis.
5. Interpret the decision in the context of the original claim: Based on whether you rejected or failed to reject the null hypothesis, state whether there is sufficient evidence to reject the claim that ACT mathematics and science scores are equal at the 0.01 significance level.

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about population parameters based on sample data. It involves formulating two competing hypotheses: the null hypothesis (H0), which represents no effect or no difference, and the alternative hypothesis (Ha), 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 H0 in favor of Ha.

Critical Value and Rejection Region

The critical value is a threshold that determines the boundary for rejecting the null hypothesis in hypothesis testing. It is derived from the significance level (α), which indicates the probability of making a Type I error. The rejection region is the range of values for the test statistic that leads to the rejection of H0; if the calculated test statistic falls within this region, we reject the null hypothesis.

Standardized Test Statistic (z)

The standardized test statistic, often denoted as z, measures how many standard deviations an observed sample mean is from the hypothesized population mean under the null hypothesis. It is calculated using the formula z = (X̄ - μ) / (σ/√n), where X̄ is the sample mean, μ is the population mean under H0, σ is the population standard deviation, and n is the sample size. This statistic is crucial for determining the position of the sample mean in relation to the null hypothesis.