91Ó°ÊÓ

The null hypothesis states the situation where there is no effect or no difference, while the alternative hypothesis claims that there is an effect or a difference. In this case, we want to determine if the population mean exceeds 2.3. Thus, Null hypothesis: \(\ H_0: \mu = 2.3\) (the population mean is equal to 2.3) Alternative hypothesis: \(\ H_1: \mu > 2.3 \) (the population mean is greater than 2.3)

The rejection region is the set of sample statistics that leads us to reject the null hypothesis, based on the significance level. In this case, we are using a 5% significance level, so our critical value will be a z-score corresponding to the value that separates the 95% of the lower area and the 5% of the upper area (right-tail) of the z-distribution. According to z-tables, that z-score is approximately 1.645 (you can find this value using tables, calculators or statistical software). Since our alternative hypothesis states that the population mean is greater than 2.3, our rejection region will be all z-scores to the right of 1.645.

The standard error of the mean (SEM) can be calculated as follows: SEM = \(\frac{s}{\sqrt{n}}\) Plugging in the given values, we have: SEM = \(\frac{0.29}{\sqrt{35}} \approx 0.049\)

Intuitively, we might think that since the sample mean is 2.4, which is greater than 2.3 the population mean might also be greater than 2.3. This is an intuition, but we need to test it statistically. To evaluate if the sample provides enough evidence to accept or reject the null hypothesis, we will use the test statistic (z-score) formula: \(z = \frac{\bar{x} - \mu_0}{\text{SEM}}\) Where \(\bar{x}\) is the sample mean, \(\mu_0\) is the value stated in the null hypothesis, and SEM is the standard error of the mean. Plugging in the given values, we get: \(z = \frac{2.4 - 2.3}{0.049} \approx 2.04\) Since our calculated z-score (2.04) is greater than the critical value (1.645), it lies in the rejection region. Therefore, we reject the null hypothesis and conclude that there is sufficient evidence to suggest that the population mean \(\mu\) is greater than 2.3.