Question

example
on a typical saturday, fiesta market reports that the mean amount of money spent by customers is $27.21 and a standard deviation of $7.93. the distribution is approximately normal.

1. what percentage of people spend less than $20? 2) what percentage of people spend above $40?
2. how many people spend between $25 and $30?

find the percentage:
lower/less than/below
what percent of students scored less than a score of x?
$$ z_{\\#}=\frac{x - \bar{x}}{s} $$
$$ p(z \leq z_{\\#}) $$
*unbounded to the left
lower: -1e99
upper: $z_{\\#}$
$\mu$: 0
$\sigma$: 1
greater/more than/above
what percent of students scored above a score of x?
$$ z_{\\#}=\frac{x - \bar{x}}{s} $$
$$ p(z \geq z_{\\#}) $$
*unbounded to the right
lower: $z_{\\#}$
upper: 1e99
$\mu$: 0
$\sigma$: 1
between
what percent of students scored between $x_{low}$ and $x_{high}$?
$$ z_{low}=\frac{x_{low}-\bar{x}}{s} $$
$$ z_{high}=\frac{x_{high}-\bar{x}}{s} $$
$$ p(z_{low} \leq z \leq z_{high}) $$
*unbounded to the left
lower: $z_{low}$
upper: $z_{high}$
$\mu$: 0
$\sigma$: 1

Explanation:

Step1: Calculate z - score for $20

Given \(\mu = 27.21\), \(\sigma=7.93\), \(x = 20\).
Using the formula \(z=\frac{x-\mu}{\sigma}\), we have \(z_1=\frac{20 - 27.21}{7.93}=\frac{-7.21}{7.93}\approx - 0.91\)
Then find \(P(Z\leq - 0.91)\) using the standard normal table. \(P(Z\leq - 0.91)=0.1814\)

Step2: Calculate z - score for $40

Given \(\mu = 27.21\), \(\sigma = 7.93\), \(x = 40\)
Using the formula \(z=\frac{x-\mu}{\sigma}\), we have \(z_2=\frac{40 - 27.21}{7.93}=\frac{12.79}{7.93}\approx1.61\)
Then find \(P(Z\geq1.61)\). Since \(P(Z\geq z)=1 - P(Z < z)\), and \(P(Z < 1.61)=0.9463\), so \(P(Z\geq1.61)=1 - 0.9463 = 0.0537\)

Step3: Calculate z - scores for $25 and $30

For \(x = 25\), \(z_{low}=\frac{25 - 27.21}{7.93}=\frac{- 2.21}{7.93}\approx - 0.28\)
For \(x = 30\), \(z_{high}=\frac{30 - 27.21}{7.93}=\frac{2.79}{7.93}\approx0.35\)
Then find \(P(-0.28\leq Z\leq0.35)\). Since \(P(-0.28\leq Z\leq0.35)=P(Z\leq0.35)-P(Z\leq - 0.28)\)
\(P(Z\leq0.35) = 0.6368\), \(P(Z\leq - 0.28)=0.3897\)
\(P(-0.28\leq Z\leq0.35)=0.6368 - 0.3897=0.2471\)

Answer:

1. Approximately \(18.14\%\) of people spend less than \(\$20\)
2. Approximately \(5.37\%\) of people spend above \(\$40\)
3. Approximately \(24.71\%\) of people spend between \(\$25\) and \(\$30\)