3. a sandwich shop is known for their fast delivery. if a delivery takes longer than 60 minutes, the order is free for the customer. the distribution of delivery times for the shop are approximately normal with a mean time of 40 minutes and a standard deviation of 8 minutes.
a. what proportion of delivery times will be 30 minutes or less?
z = (30 - 40)/8=-1.25 z

Question

3. a sandwich shop is known for their fast delivery. if a delivery takes longer than 60 minutes, the order is free for the customer. the distribution of delivery times for the shop are approximately normal with a mean time of 40 minutes and a standard deviation of 8 minutes.
a. what proportion of delivery times will be 30 minutes or less?
z = (30 - 40)/8=-1.25  z <.1056
b. what proportion of the deliveries are expected to be free because of the 60 - minute time limit?
z=(60 - 40)/8 = 2.5  .9938
c. the manager of the sandwich shop needs to predict the expected total cost for the free deliveries for the next month. based on previous months, the manager knows that the shop will have about 600 deliveries and the average delivery order cost is $22. how much should the manager expect the free deliveries to cost? show your work.
4. the mean home value in atlanta, georgia is $386,122 with a standard deviation of $221,000. a small home valued at $120,000 is at the 20th percentile.
a. interpret the percentile for the small home.
b. calculate and interpret the z - score for the small home.
c. is it reasonable to assume that the distribution of atlanta home values is approximately normal? explain.
5. the distribution of final exam scores in a college chemistry class is skewed to the left with a mean of 88 and a standard deviation of 12.
a. calculate and interpret the z - score for paige who scored a 95 on the exam.
b. if possible, calculate the proportion of students in this class that scored less than paige.

Accepted Answer

### 3.
#### a.
# Explanation:
## Step1: Calculate z - score
The formula for the z - score is $z=\frac{x-\mu}{\sigma}$, where $x = 30$ minutes, $\mu=40$ minutes and $\sigma = 8$ minutes. So, $z=\frac{30 - 40}{8}=\frac{- 10}{8}=-1.25$.
## Step2: Find proportion from z - table
Looking up the z - score of $-1.25$ in the standard normal distribution table, the proportion of values to the left of $z=-1.25$ is $0.1056$.
# Answer:
$0.1056$

#### b.
# Explanation:
## Step1: Calculate z - score
For $x = 60$ minutes, $\mu = 40$ minutes and $\sigma=8$ minutes. Using the z - score formula $z=\frac{x-\mu}{\sigma}$, we get $z=\frac{60 - 40}{8}=\frac{20}{8}=2.5$.
## Step2: Find proportion from z - table
The proportion of values to the right of $z = 2.5$ is $1 - P(Z\leq2.5)$. Looking up $P(Z\leq2.5)$ in the standard - normal table, $P(Z\leq2.5)=0.9938$, so the proportion of values to the right is $1 - 0.9938=0.0062$.
# Answer:
$0.0062$

#### c.
# Explanation:
## Step1: Identify proportion of free deliveries
From part (b), the proportion of free deliveries is $p = 0.0062$.
## Step2: Calculate number of free deliveries
The number of deliveries $n = 600$. The number of free deliveries is $n	imes p=600	imes0.0062 = 3.72$.
## Step3: Calculate expected cost of free deliveries
The average cost per delivery is $\$22$. The expected cost of free deliveries is $3.72	imes22=\$81.84$.
# Answer:
$\$81.84$

### 4.
#### a.
# Brief Explanations:
The 20th percentile means that 20% of the home values in Atlanta are less than the value of the small home ($\$120,000$), and 80% of the home values are greater than $\$120,000$.
# Answer:
20% of Atlanta home values are less than $\$120,000$ and 80% are greater.

#### b.
# Explanation:
## Step1: Calculate z - score
The z - score formula is $z=\frac{x-\mu}{\sigma}$, where $x = 120000$, $\mu = 386122$ and $\sigma=221000$. So, $z=\frac{120000 - 386122}{221000}=\frac{-266122}{221000}\approx - 1.20$.
## Step2: Interpret z - score
A z - score of approximately $-1.20$ means that the value of the small home is approximately $1.20$ standard deviations below the mean home value in Atlanta.
# Answer:
$z\approx - 1.20$. The home value is about 1.20 standard deviations below the mean.

#### c.
# Brief Explanations:
It may not be reasonable to assume a normal distribution. Home values are often skewed right because there are a few very high - value homes (outliers) that can pull the mean upwards. Also, there is a natural lower bound of 0 for home values but no natural upper bound, which can cause skewness.
# Answer:
No. Home values often have outliers and a natural lower bound, which can cause right - skewness.

### 5.
#### a.
# Explanation:
## Step1: Calculate z - score
The z - score formula is $z=\frac{x-\mu}{\sigma}$, where $x = 95$, $\mu = 88$ and $\sigma = 12$. So, $z=\frac{95 - 88}{12}=\frac{7}{12}\approx0.58$.
## Step2: Interpret z - score
A z - score of approximately $0.58$ means that Paige's score is approximately $0.58$ standard deviations above the mean score of the class.
# Answer:
$z\approx0.58$. Paige's score is about 0.58 standard deviations above the mean.

#### b.
# Brief Explanations:
Since the distribution is skewed to the left, we cannot use the standard normal distribution (z - table) to find the proportion of students who scored less than Paige. We would need more information about the specific non - normal distribution (such as the shape of the skewed distribution or more percentiles) to calculate this proportion.
# Answer:
Not possible. The distribution is skewed, so we can't use the normal distribution methods.

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QUESTION IMAGE

Question

Explanation:

3.

a.

Step1: Calculate z - score

Step2: Find proportion from z - table

Step1: Calculate z - score

Step2: Find proportion from z - table

Step1: Identify proportion of free deliveries

Step2: Calculate number of free deliveries

Step3: Calculate expected cost of free deliveries

Answer:

b.