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Hypothesis testing is a method used in statistics to make inferences about a population based on sample data. In this exercise, we are testing whether this year's college entrants have higher IQs than last year's entrants. This involves setting up two opposing statements:

• The null hypothesis (\(H_0\)) assumes no difference in the average IQs between the two groups. Mathematically, it's written as \(\mu_1 = \mu_2\).
• The alternative hypothesis (\(H_a\)) suggests that this year's students have a higher average IQ, which is expressed as \(\mu_1 > \mu_2\).

Establishing these hypotheses helps determine the direction and nature of our test. By analyzing sample statistics, we can decide whether there's enough evidence to support the alternative hypothesis.

The normal distribution is a probability distribution that is symmetric and bell-shaped, describing how the values of a variable are distributed. Many real-world phenomena, including IQ scores, approximate a normal distribution. When performing hypothesis tests for means, assuming normal distribution is key because:

• It simplifies calculations since test statistics often rely on the properties of normal or nearly normal distributions.
• It allows us to apply the Central Limit Theorem, which states that the distribution of sample means tends to be normal, especially with larger sample sizes.

In this exercise, we assume the IQ scores in both years are normally distributed and have equal variances, making it a perfect scenario for using a t-test.

The level of significance, denoted by \(\alpha\), is the threshold we set to determine whether a statistical result is considered significant. It reflects the probability of rejecting the null hypothesis when it is true, known as a Type I error. In this exercise, we use a \(5\%\) level of significance (\(\alpha = 0.05\)). This means we are willing to accept a \(5\%\) risk of stating that this year's students have higher IQs when there actually is no difference.During hypothesis testing, the test statistic is compared to a critical value, which depends on \(\alpha\). If the test statistic falls beyond this critical value in the direction of the alternative hypothesis, we reject \(H_0\). Here, the calculated \(t\)-value is \(2.30\), which exceeds the critical value of \(1.943\) at \(df = 6\), leading to rejection of the null hypothesis.

Sample statistics, such as the mean and standard deviation, provide concrete numerical summaries of the sample data. In hypothesis testing, these statistics are vital in calculating the test statistic.From this year's student sample, the mean IQ is \(114\) and the standard deviation is \(3.51\). Whereas, for last year's sample, the mean IQ is \(111\), with a standard deviation of \(1.54\). These summary statistics feed into the formula for calculating the t-test statistic:\[ t = \frac{(\bar{x}_1 - \bar{x}_2)}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}} \]This formula provides a standardized way to compare the means from two samples, considering sample size and variability. Understanding these statistics helps in determining the validity of the hypothesis test.