6. Ten students take a test and the scores are {80, 99, 83, 82, 100, 75, 83, 85,

Question

6. Ten students take a test and the scores are {80, 99, 83, 82, 100, 75, 83, 85, 71, 92}. (a) Compute the sample mean and sample standard deviation. (b) Let X represent the normal random variable whose mean is the sample mean and whose standard deviation is the sample standard deviation. What is P (X > 90)? What proportion of students actually did score above a 90? (c) If an A is given to any student whose score is greater than 1 standard deviation above the mean (using normal with sample mean and sample deviation), how many students get an A? (d) If a normal random variable with mean 80 and standard deviation 10 is used, how many students get an A (more than 1 standard deviation away from the mean)?

Explanation / Answer

a)mean=85

standard deviation =9.475114

b)x follows normal distribution with mean 85 and variance 89.778.

p[x>90]=p[{(x-85)/9.475}>(5/9.475)]=p[z>.53]=1-p[z<.53]=0.2981 ,where z={(x-85)/9.475) follows standard noral distribution.

here 29.81% student did 90 above score.

actually 30% student did 90 above score.

c)1 standard deviation above mean=9.475+85=94.475

so 2 students will get A.

d)if mean 80 and standard deviation 10

then 1 standard deviation away from mean=10+80=90

so now 3 students will get A

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