Final answer:
To test if the students in the Department of Economics have above average intelligence, a hypothesis test is set up with the null hypothesis that the mean IQ is 100 and the alternative hypothesis that the mean IQ is greater than 100. A z-test is appropriate as the sample size is large and the population standard deviation is known. We look for a p-value less than the level of significance to support the claim of above-average intelligence.
Step-by-step explanation:
To determine whether there is sufficient evidence to support the head's claim that the students in the Department of Economics have above average intelligence, we can set up a hypothesis test. We know the mean population IQ is 100 with a standard deviation of 15. Given a sample of thirty students with a mean score of 112.5, we can formulate the null hypothesis (H0) and the alternative hypothesis (H1) as follows:
- H0: μ = 100 (The mean IQ of the students is equal to the general population mean IQ)
- H1: μ > 100 (The mean IQ of the students is greater than the general population mean IQ, indicating above-average intelligence)
The appropriate test statistic for this hypothesis test is the z-test because the population standard deviation is known and the sample size is large (n > 30).
To perform a z-test, we calculate the z-score using the sample mean, population mean, standard deviation, and sample size. Based on this z-score, we then look up the p-value in a standard normal distribution table. If the p-value is less than the level of significance (usually set at 0.05), we reject the null hypothesis in favor of the alternative hypothesis. If the calculated p-value is greater, there is insufficient evidence to claim that the students have above-average intelligence.