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 English and Reading Scores The mean ACT English score for 120 high school students is 19.9. Assume the population standard deviation is 7.2. The mean ACT reading score for 150 high school students is 21.2. Assume the population standard deviation is 7.1. At α=0.10, can you support the claim that ACT reading scores are higher than ACT English scores? (Source: ACT, Inc.)

Verified step by step guidance

1. Step 1: Identify the claim and state the null hypothesis (Ho) and alternative hypothesis (Ha). The claim is that ACT reading scores are higher than ACT English scores. This translates to Ha: μ_reading > μ_english (alternative hypothesis) and Ho: μ_reading ≤ μ_english (null hypothesis).
2. Step 2: Determine the critical value(s) and rejection region(s). Since the significance level α = 0.10 and the test is one-tailed (greater than), use a z-table to find the critical value corresponding to α = 0.10. The rejection region is z > critical value.
3. Step 3: Calculate the standardized test statistic z. Use the formula: $z=\frac{(\mu }{}$_reading-μ_english)σ_reading2n_reading+σ_english2n_english. Plug in the values: μ_reading = 21.2, μ_english = 19.9, σ_reading = 7.1, σ_english = 7.2, n_reading = 150, n_english = 120.
4. Step 4: Compare the calculated z-value to the critical value. If the z-value falls in the rejection region (z > critical value), reject the null hypothesis. Otherwise, fail to reject the null hypothesis.
5. Step 5: Interpret the decision in the context of the original claim. If the null hypothesis is rejected, conclude that there is sufficient evidence to support the claim that ACT reading scores are higher than ACT English scores. If the null hypothesis is not rejected, conclude that there is insufficient evidence to support the claim.

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 represents the effect or difference we suspect. In this context, we are testing whether the mean ACT reading score is significantly higher than the mean ACT English score.

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 rejecting H0. In this case, with α=0.10, we will find the critical z-value to establish the rejection region for the test.

Standardized Test Statistic (z)

The standardized test statistic, often denoted as z, measures how many standard deviations an element is from the mean. It is calculated using the sample means, population standard deviations, and sample sizes. This statistic allows us to compare the observed difference between the sample means to what we would expect under the null hypothesis, facilitating the decision to reject or fail to reject H0 based on its value relative to the critical value.