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In 2007, a study of nearly 250,000 18-19-year-old Norwegian males found that first-borns have an IQ that is 2.3 points higher than those who are second-born. To determine if similar evidence exists at your university, you collect data from 250 students, of which 140 are first-borns. After administering an IQ test, you find that the first-borns score an average of 108.3 with a standard deviation of 13.2, while the second-borns score an average of 107.1 with a standard deviation of 11.6.

You hypothesize that first-borns and second-borns in a university population have identical IQs, against the one-sided alternative hypothesis that first-borns have higher IQs. Using a significance level of 5%, what is your conclusion?

Answer :

Based on the given data and using a significance level of 5%, we do not find evidence to support the claim that first-borns have higher IQs than second-borns in this university population.

Based on the information provided, we can conduct a hypothesis test to determine if first-borns and second-borns in a university population have identical IQs.
First, we need to state our null and alternative hypotheses. The null hypothesis (H0) is that there is no difference in the mean IQ scores of first-borns and second-borns. The alternative hypothesis (Ha) is that first-borns have higher IQs than second-borns.

Next, we calculate the test statistic. We can use a t-test because the sample sizes are relatively small and we don't know the population standard deviations. The formula for the t-test is:
t = (mean1 - mean2) / √((s1² / n1) + (s2² / n2))
In this case, mean1 is the mean IQ score of first-borns (108.3), mean2 is the mean IQ score of second-borns (107.1), s1 is the standard deviation of first-borns (13.2), s2 is the standard deviation of second-borns (11.6), n1 is the number of first-borns (140), and n2 is the number of second-borns (110).

After calculating the test statistic, we compare it to the critical value from the t-distribution table. With a significance level (alpha) of 0.05 and degrees of freedom equal to the smaller sample size minus 1, we find the critical value. If the test statistic is greater than the critical value, we reject the null hypothesis in favor of the alternative hypothesis.
Based on the calculations, the test statistic is approximately 0.625 and the critical value is approximately 1.664. Since the test statistic is not greater than the critical value, we fail to reject the null hypothesis. Therefore, we do not have enough evidence to conclude that first-borns have higher IQs than second-borns in this university population.
In conclusion, based on the given data and using a significance level of 5%, we do not find evidence to support the claim that first-borns have higher IQs than second-borns in this university population.

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