Question

Scores on a university exam are Normally distributed with a mean of 75 and a standard deviation of 5. The professor teaching the class declares that a score of 80 or higher is required for a grade of at least "B." Using the 68-95-99.7 rule, what percent of students failed to earn a grade of at least "B"? 5% 84% 75% 16%

          Scores on a university exam are Normally distributed with a mean of 75 and a standard deviation of 5. The professor teaching the class declares that a score of 80 or higher is required for a grade of at least "B." Using the 68-95-99.7 rule, what percent of students failed to earn a grade of at least "B"?

5%
84%
75%
16%

Show more…

Scores on a university exam are Normally distributed with a mean of 75 and a standard deviation of 5. The professor teaching the class declares that a score of 80 or higher is required for a grade of at least "B." Using the 68-95-99.7 rule, what percent of students failed to earn a grade of at least "B"?

5%
84%
75%
16%

Added by Leah Y.

Elementary Statistics a Step by Step Approach

Allan G. Bluman 9th Edition

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Solved by Expert Sanchit Jain

03/31/2023

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Scores on a university exam are normally distributed with a mean of 75 and a standard deviation of 5. The professor teaching the class declares that a score of 80 or higher is required for a grade of at least "B." Using the 68-95-99.7 rule, what percent of students failed to earn a grade of at least "B"?

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Transcript

00:01 Hello students we are going to write here in this question our mean value is given as 75 and standard aviation is given as sigma we are going to apply here empirical rule so we need to write here for this what would be our empirical rule we need to write here for this so first of all we are going to use a normal distribution within the question and if i write here a according to our normal distribution, if i write here all the random variables are distributed amongst bell -shaped curve.

00:40 So we can understand our empirical rule by our craft.

00:44 If i write here, let's say this is probability of pz.

00:47 So definitely this is z equals to 0 or x is equal to mu.

00:51 So if i write here for this, let's say this is x equals to 1 and this is minus 1.

00:56 We need to write here, this is minus 2 and this is plus 2.

01:01 Or to make more clear we can use other pens so this is let's say 1 minus 1 1 2 and minus 2 this is minus 3 so basically if i write here for this total area and a normal distribution curve is 1 so basically in x terms we can say we can see if i write here jd score attribute is xx minus mu divided by sigma so definitely x value is j sigma j sigma plus mu and if i write here for this this must be so if j is positive z is positive then this is mu plus z sigma if z is negative then mu minus z sigma so x1 is x1 is mu plus z sigma and x2 is mu minus z sigma so basically if i write here for this for empirical rule we can have three conditions right here so here for this our x values are probability of jad values between minus one to one that is equivalent to probability of x values between mu minus sigma if i right here for this mu minus sigma and mu plus sigma that must be equivalent to 68 and half of this is 34%.

02:37 So let's say this is our 34 % area and this total area is 68%...

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