Question

At the beginning of school year the average reading comprehension of freshmen college students was recorded at 79.40. After 3 months of studying the same instruments for reading comprehension test was used for 56 students and scored an average of 86.48 with a standard deviation of 4.65. Does staying in the college improve reading comprehension of the freshmen students? Use 0.05 level of significance

Answer 1

$a_0=79.4\\ n=56\\ \overline{x}=86.48\\ s=4.65\\ \alpha=0.05\\ H_0: \mu=a_0=79.4, H_1:\mu>a_0=79.4\\ \mu\text{ --- reading comprehension of the students after}\\ \text{3 months of studying}.\\ \text{We assume that reading comprehension of the students}\\ \text{has normal distribution}.\\ \text{We will use the following random variable as a criterion:}\\ T=\frac{(\overline{X}-a_0)\sqrt{n}}{s}\\ T\text{ has t-distribution with } k=n-1 \text{ degrees of freedom}.\\ \text{Observed value:}\\ t_{obs}=\frac{(86.48-79.4)\sqrt{56}}{4.65}\approx 11.4\\ \text{Critical value (one-sided)}:\\ t_{cr}=t_{cr}(\alpha;k)=t_{cr}(0.05;55)\approx 1.673\\ (1.673,\infty)\text{ --- critical region}\\ t_{obs}\text{ falls into the critical region. So we reject } H_0.\\ \text{Staying in the college improves reading comprehension}\\ \text{of the freshmen students}.$