A parent interest group is looking at whether birth order affects scores on the ACT test. It was suggested that, on average, first-born children earn lower ACT scores than second-born children. After surveying a random sample of 175 first-born children, the parents' group found that they had a mean score of 24.2 on the ACT. A survey of 100 second-born children resulted in a mean ACT score of 24.5. Assume that the population standard deviation for first-born children is known to be 1.7 points and the population standard deviation for second-born children is known to be 1.1 points. Is there sufficient evidence at the 1 \% level of significance to say that the mean ACT score of first-born children is lower than the mean ACT score of second-born children? Let first-born children be Population 1 and let second-born children be Population 2. Step 3 of 3: Draw a conclusion and interpret the decision. Answer We fail to reject the null hypothesis and conclude that there is insufficient evidence at a 0.01 level of significance to support the parent interest group's claim that firstborn children earn lower ACT scores on average than second-born children. We reject the null hypothesis and conclude that there is sufficient evidence at a 0.01 level of significance to support the parent interest group's claim that first-born children earn lower ACT scores on average than second-born children. We fail to reject the null hypothesis and conclude that there is sufficient evidence at a 0.01 level of significance to support the parent interest group's claim that firstborn children earn lower ACT scores on average than second-born children. We reject the null hypothesis and conclude that there is insufficient evidence at a 0.01 level of significance to support the parent interest group's claim that first-born children earn lower ACT scores on average than second-born children. Submit Answer
Solution
Step 1 ：Define the null hypothesis as the mean ACT score of first-born children being equal to the mean ACT score of second-born children. The alternative hypothesis is that the mean ACT score of first-born children is lower than the mean ACT score of second-born children.
Step 2 ：Given the sample sizes, sample means, and population standard deviations for both groups, calculate the test statistic and the p-value. The sample size for first-born children (n1) is 175, the mean ACT score (x1) is 24.2, and the population standard deviation (s1) is 1.7. The sample size for second-born children (n2) is 100, the mean ACT score (x2) is 24.5, and the population standard deviation (s2) is 1.1.
Step 3 ：Calculate the test statistic (z) and the p-value. The calculated z-value is -1.7734943126169553 and the p-value is 0.0380734167386726.
Step 4 ：Compare the p-value with the level of significance (0.01). If the p-value is less than the level of significance, reject the null hypothesis and conclude that there is sufficient evidence to support the claim that first-born children earn lower ACT scores on average than second-born children. If the p-value is greater than the level of significance, fail to reject the null hypothesis and conclude that there is insufficient evidence to support the claim.
Step 5 ：Since the p-value (0.0380734167386726) is greater than the level of significance (0.01), we fail to reject the null hypothesis. This means that there is insufficient evidence at a 0.01 level of significance to support the parent interest group's claim that first-born children earn lower ACT scores on average than second-born children.
Step 6 ：\(\boxed{\text{Final Answer: We fail to reject the null hypothesis and conclude that there is insufficient evidence at a 0.01 level of significance to support the parent interest group's claim that first-born children earn lower ACT scores on average than second-born children.}}\)
