Solve the following problem: 1. The salaries of 30 teachers in a small school are normally distributed with a mean of P24,300 with a standard deviation of P4, 365. Assuming normality of the data, how many of the teachers (a) receive a salary of P20,000 and below? (b) receive a salary above P30,000? 2. The average temperature recorded in the city for the month of January this year is 17.13°C with a standard deviation of 2.23°C. Assuming a normally distributed data, find the probability that January next year, the city will have a temperature (a) below 13°C. (b) more than 15°C but not over 19°C. 3. The average score of a class in a statistics examination is 65 out of 100 with a standard deviation of 10. What percent of students have a score below 75?
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Answers with step-by-step explanations:
1. Let X be the salary of a teacher.
(a) We need to find P(X ≤ 20,000). Using the z-score formula, we have:
z = (20,000 - 24,300) / 4,365 = -0.98
Looking up the standard normal distribution table, we find that P(Z ≤ -0.98) = 0.1635. Therefore, about 16.35% of the teachers receive a salary of P20,000 and below.
(b) We need to find P(X > 30,000). Again, using the z-score formula, we have:
z = (30,000 - 24,300) / 4,365 = 1.31
Looking up the standard normal distribution table, we find that P(Z > 1.31) = 0.0951. Therefore, about 9.51% of the teachers receive a salary above P30,000.
2. Let X be the temperature recorded in January next year.
(a) We need to find P(X < 13). Using the z-score formula, we have:
z = (13 - 17.13) / 2.23 = -1.85
Looking up the standard normal distribution table, we find that P(Z < -1.85) = 0.0322. Therefore, the probability that the temperature in January next year will be below 13°C is about 3.22%.
(b) We need to find P(15 < X < 19). Again, using the z-score formula, we have:
z1 = (15 - 17.13) / 2.23 = -0.95
z2 = (19 - 17.13) / 2.23 = 0.84
Using the standard normal distribution table, we find that P(-0.95 < Z < 0.84) = P(Z < 0.84) - P(Z < -0.95) = 0.7995 - 0.1711 = 0.6284. Therefore, the probability that the temperature in January next year will be more than 15°C but not over 19°C is about 62.84%.
3. Let X be the score of a student in the statistics examination. We need to find P(X < 75).
Using the z-score formula, we have:
z = (75 - 65) / 10 = 1
Using the standard normal distribution table, we find that P(Z < 1) = 0.8413. Therefore, about 84.13% of the students have a score below 75.