Answer to Question #155106 in Statistics and Probability for Darshini

Question #155106

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


1
Expert's answer
2021-01-12T16:48:16-0500

The means score for students is


μ=4560=0.75\mu =\frac{45}{60} =0.75

The standard deviation is


σ=1260=0.20\sigma = \frac{12}{60} = 0.20

The number of students


n=200n=200

The Z value is obtained as:



Z=xμσZ=\frac{x-\mu}{\sigma}

Let xlx_l Be the score for a student with 24


xl=2460=0.40x_l= \frac{24}{60}=0.40

The corresponding z score is



Z=0.400.750.20=1.7500Z=\frac{0.40-0.75}{0.20}= -1.7500

The probability that


P(Z1.75)=0.040P(Z \le-1.75) = 0.040


Let xux_u Be the score for a student with 54

xu=5460=0.90x_u =\frac{54}{60}= 0.90

The corresponding Z value is



Z=0.900.750.20=0.75Z=\frac{0.90-0.75}{0.20}= 0.75

The probability that


P(Z0.75)=0.773P(Z \le 0.75) = 0.773

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


0.7730.040=0.7330.773-0.040=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%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%22.7\% of the students scored above 54 out of 60 marks.


The number of students that scored between 24 and 54 was



2000.733=146.6147 students200*0.733=146.6 \approx147\ students


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