Question

A teacher looks at the scores on a standardized test, where the mean of the test was a 73 and the standard deviation was 9. Find the following

z-score table Link

1. What is the probability that a student will score between 73 and 90
2. What is the probability that a student will score between 65 and 85
3. What is the probability that a student will get a score greater than 70
4. What is the probability that a student will score more than 80
5. What score would give you the top 12.1%

Answer 1

Given :

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The probability that a student will score between 73 and 90 = P(73<x<90)

First find the z-scores for x=73 and x=90

For x=73

For x=90

Required probability is P( 0 < z < 1.89 )

= P( z < 1.89 ) - P( z < 0 )

= 0.9706 - 0.5000          ( From the z-score table for z=1.89 and for z =0)

= 0.4706

The probability that a student will score between 73 and 90 is 0.4706

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the probability that a student will score between 65 and 85 = P(65<x<85)

First find the z-scores for x=65 and x=85

For x=65

For x=85

Required probability is P( -0.89 < z < 1.33 )

= P( z < 1.33 ) - P( z < -0.89 )

= 0.9082 - 0.1867                  ( From the z-score table for z=1.33 and for z = -0.89 )

= 0.7215

The probability that a student will score between 73 and 90 is 0.7215

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the probability that a student will get a score greater than 70 = P(x>70)

using the z-score formula

here x=70

Required probability is P(z>-0.33) = P(z<0.33)

= 0.6293        ( From z-score table for z=0.33)

The probability that a student will get a score greater than 70 is 0.6293

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the probability that a student will score more than 80 = P(x>80)

using the z-score formula

here x=80

Required probability is P(z>0.78) = 1 - P(z<0.78)

= 1 - 0.7823           ( From z-score table for z=0.78 )

= 0.2177

The probability that a student will score more than 80 is 0.2177

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score would give you the top 12.1%

P(Z>z) = 0.1210

P(Z<z) = 1 - 0.1210

P(Z<z) = 0.8790

Now looking in body (middle part ) of z-score table for area closest to 0.8790 then note down the corresponding z value.

z = 1.17

Now use z-score formula to get the value of x ( required score)

The score that would give you the top 12.1% is 83.53

Rounded to nearest whole number would be 84