Question

fill in the missing information. tim worker is doing his budget. he discovers that the average miscellaneous expense is $45.00 with a standard deviation of $16.00. what percent of his expense in this category would he expect to fall between $38.60 and $57.80? the z for $38.60 = the percent of area associated with $38.60 = % the z for $57.80 = the percent of area associated with $57.80 = % adding the two together, the percent of his expenses between $38.60 and $57.80 is %

Explanation:

Step1: Calculate z - score for $38.60

The formula for the z - score is $z=\frac{x - \mu}{\sigma}$, where $\mu$ is the mean, $\sigma$ is the standard deviation, and $x$ is the value. Given $\mu = 45$, $\sigma=16$, and $x = 38.60$.
$z_1=\frac{38.60 - 45}{16}=\frac{- 6.4}{16}=-0.4$

Step2: Find the area associated with $z = - 0.4$

Looking up in the standard - normal table, the area to the left of $z=-0.4$ is $0.3446$ or $34.46\%$.

Step3: Calculate z - score for $57.80

Using the z - score formula with $x = 57.80$, $\mu = 45$, and $\sigma = 16$.
$z_2=\frac{57.80 - 45}{16}=\frac{12.8}{16}=0.8$

Step4: Find the area associated with $z = 0.8$

Looking up in the standard - normal table, the area to the left of $z = 0.8$ is $0.7881$ or $78.81\%$.

Step5: Calculate the area between $z=-0.4$ and $z = 0.8$

The area between the two z - scores is $0.7881-0.3446 = 0.4435$ or $44.35\%$.

Answer:

The z for $38.60=-0.4$
The percent of area associated with $38.60 = 34.46\%$
The z for $57.80 = 0.8$
The percent of area associated with $57.80 = 78.81\%$
Adding the two together, the percent of his expenses between $38.60$ and $57.80$ is $44.35\%$