Problem
A math teacher claims that she has developed a review course that increases the scores of students on the math portion of a college entrance exam. Based on data from the administrator of the exam, scores are normally distributed with $\mu=521$. The teacher obtains a random sample of 1800 students, puts them through the review class, and finds that the mean math score of the 1800 students is 528 with a standard deviation of 116 . Complete parts (a) through (d) below. Find the P-value. The P-value is 0.005 (Round to three decimal places as needed.) Is the sample mean statistically significantly higher? A. No, because the $\mathrm{P}$-value is less than $\alpha=0.10$ B. Yes, because the P-value is greater than $\alpha=0.10$. C. Yes, because the P-value is less than $\alpha=0.10$ D. No, because the P-value is greater than $\alpha=0.10$.
Solution
Step 1 ：Calculate the standard error using the formula \( \text{standard error} = \frac{\text{sample standard deviation}}{\sqrt{\text{sample size}}} \)
Step 2 ：Calculate the z-score using the formula \( z = \frac{\text{sample mean} - \text{population mean}}{\text{standard error}} \)
Step 3 ：Find the P-value corresponding to the calculated z-score
Step 4 ：Determine if the sample mean is statistically significantly higher by comparing the P-value to \( \alpha = 0.10 \)
Step 5 ：\( \text{Final Answer:} \boxed{0.005} \)
