Answer to Question #155106 in Statistics and Probability for Darshini

Question

It has been observed that in Fundamental Statistics course, a students receves on average of 45 marks with a standard deviation of 12 marks out of 60. If the marks obtained students normally distributed Then • Find the probability that a student will score the marks between 24 and 54 Interpret your answer if there are 200 students in the class. How many students we can expect to score marks between 24 to 54

Answer 1

The means score for students is

$$\mu =\frac{45}{60} =0.75$$

The standard deviation is

$$\sigma = \frac{12}{60} = 0.20$$

The number of students

$$n=200$$

The Z value is obtained as:

$$Z=\frac{x-\mu}{\sigma}$$

Let $x_l$ Be the score for a student with 24

$$x_l= \frac{24}{60}=0.40$$

The corresponding z score is

$$Z=\frac{0.40-0.75}{0.20}= -1.7500$$

The probability that

$$P(Z \le-1.75) = 0.040$$

Let $x_u$ Be the score for a student with 54

$$x_u =\frac{54}{60}= 0.90$$

The corresponding Z value is

$$Z=\frac{0.90-0.75}{0.20}= 0.75$$

The probability that

$$P(Z \le 0.75) = 0.773$$

The probability that a student scores between 24 and 54 out of 60 is

$$0.773-0.040=0.733$$

We use the normal distribution to estimate the probability of students scoring between 24 and 54 marks. The probability corresponding to a score of 24 is 0.040 indicating that $4\%$ of students scored below 24 out of 60. Similarly a probability of 0.773 corresponding to a score of 54 indicates that $22.7\%$ of the students scored above 54 out of 60 marks.

The number of students that scored between 24 and 54 was

$$200*0.733=146.6 \approx147\ students$$