According to the IRS,2012 tax returns showed an average refund of $2995 with a s

Question

According to the IRS,2012 tax returns showed an average refund of $2995 with a standard deviation of $2607.Assume that 2012 tax retirn are normally distributed. round answer to 4 decimals.

A) Find the probability a randomly selected 2012 tax return showed an amount no less than $2,842 and at most $5,802. B)Find the probability a randomly selected 2012 tax return showed an amount greater than $3,185. C)Find the probability a randomly selected 2012 tax return showed an amount less than or equal to $4,048 or greater than $5,008. D) Find the probability a randomly selected 2012 tax return showed an amount less than $189. E)The probability is 0.15 that a randomly selected 2012 tax return showed no more than what amount? (Remember the label.) F)The probability is 0.97 that a randomly selected 2012 tax return showed at least what amount? (Remember the label.)

Explanation / Answer

mean= 2995 , s = 2607

a)
P( 2842 < x < 5802)

z = (( 2842 - 2995) / 2607 <z < ( 5802 - 2995)/ 2607)
= (-0.058 < z < 1.076) by using standard table

P( 2842 < x < 5802) = (-0.058 <z < 1.076)
= 0.3826
b)
P(x> 3185)

z = ( 3185- 2995) / 2607
= 0.073

we need to find P(z > 0.073) by standard table

P(x> 3185) = P(z > 0.073) = 0.471

c)

P( 4048 < x < 5008)

z = (( 4048 - 2995) / 2607 <z < ( 5008 - 2995)/ 2607)
= (0.4039 < z < 0.7721) by using standard table

P( 4048 < x < 5008) = (0.4039 <z < 0.7721)
= 0.1231

d)
P(x <189)

z = (189 - 2995) / 2607
= -1.0763

we need to find P(z < -1.0763) by standard table

P(x < 189) = P(z < -1.0763) = 0.1409

e)
z value for 0.15 = -1.0364
z = ( x - mean) / s
-1.0364 = ( x - 2995) / 2607
x = 293.1052

f)
z value for 0.15 = -1.0364
z = ( x - mean) / s
1.8807 = ( x - 2995) / 2607
x = 7897.9849

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