Question 1

A bottler of drinking water fills plastic bottles with a mean volume of 1001 milliliters (mL) and standard deviation 6 mL. The fill volumes are normally distributed. What
Proportion of bottles have volumes less than 1001 mL?

Accepted Answer

In a normal distribution, the mean divides the distribution in half. Therefore, 50% of the values lie below the mean. Since the mean volume is 1001 mL, 50% of the bottles have volumes less than 1001 mL.

Question 2

The average salary of a male full professor at a public four-year institution offering classes at the doctoral level is $99,685. For a female full professor at the same kind of
Institution, the salary is $90,330. If the standard deviation for the salaries of both genders
Is approximately $5200 and the salaries are normally distributed, find the 80th percentile
Salary for the male professors and for female professors, respectively.

Accepted Answer

To find the 80th percentile salary for both male and female professors, we use the normal distribution properties. The 80th percentile corresponds to a Z-score of approximately 0.84 (from Z-tables or calculators). The formula to calculate a specific value given a Z-score is X = μ + Zσ, where μ is the mean, σ is the standard deviation, and Z is the Z-score. For male professors: $99,685 + 0.84 * $5,200 ≈ $104,053. For female professors: $90,330 + 0.84 * $5,200 ≈ $94,698.

Question 3

If a normally distributed group of test scores have a mean of 70 and a standard deviation of 12, find the percentage of scores that will fall below 50.

Accepted Answer

The answer of If a normally distributed group of test...

Question 4

A normal population has a mean $\mu = 40$ and standard deviation $\sigma = 11$. What proportion of the population is between 24 and $32 ?$

Accepted Answer

To find the proportion of the population between 24 and 32, we use the Z-score formula: $Z = \frac{X - \mu}{\sigma}$. For X = 24, $Z = \frac{24 - 40}{11} = -1.45$. For X = 32, $Z = \frac{32 - 40}{11} = -0.73$. Looking these Z-scores up in a standard normal distribution table or using a calculator, we find the area (proportion of the population) between these Z-scores is approximately 0.1592.

Question 5

A normal population has a mean $\mu = 33$ and standard deviation $\sigma = 9$. What is the probability that a randomly chosen value will be greater than 44 ?

Accepted Answer

To find the probability that a randomly chosen value from a normal distribution is greater than a specific value, we use the Z-score formula: $Z = \frac{X - \mu}{\sigma}$, where $X$ is the value in question, $\mu$ is the mean, and $\sigma$ is the standard deviation. For $X = 44$, $\mu = 33$, and $\sigma = 9$, the Z-score is $Z = \frac{44 - 33}{9} = \frac{11}{9} \approx 1.22$. Looking up this Z-score in a standard normal distribution table or using a calculator, we find the probability of being less than 44 is approximately 0.8888. Therefore, the probability of being greater than 44 is $1 - 0.8888 = 0.1112$.

Question 6

A bottler of drinking water fills plastic bottles with a mean volume of 999 milliliters (mL) and standard deviation 5 mL. The fill volumes are normally distributed. What
Proportion of bottles have volumes between 992 mL and 998 mL?

Accepted Answer

The answer of A bottler of drinking water fills plastic...

Question 7

The average hourly wage of workers at a fast food restaurant is $6.50/hr. Assume the wages are normally distributed with a standard deviation of $0.45. If a worker at this fast
Food restaurant is selected at random, what is the probability that the worker earns more
Than $6.75?

Accepted Answer

To find the probability that a worker earns more than $6.75, we use the Z-score formula: $Z = \frac{X - \mu}{\sigma}$, where $X$ is the value of interest ($6.75), $\mu$ is the mean ($6.50), and $\sigma$ is the standard deviation ($0.45). Plugging in the values, we get $Z = \frac{6.75 - 6.50}{0.45} = \frac{0.25}{0.45} \approx 0.56$. Looking up 0.56 on the standard normal distribution table gives us the area to the left of 0.56, which is approximately 0.7123. To find the probability of earning more than $6.75, we subtract this value from 1 (since the total area under the curve is 1). Thus, $1 - 0.7123 = 0.2877$, or 28.8%.

Question 8

The average length of crocodiles in a swamp is 11.5 feet. If the lengths are normally distributed with a standard deviation of 1.7 feet, find the probability that a crocodile is
More than 11 feet long.

Accepted Answer

The answer of The average length of crocodiles in a...

Question 9

At a large department store, the average number of years of employment for a cashier is 5.7 with a standard deviation of 1.8 years, and the distribution is approximately normal.
If an employee is picked at random, what is the probability that the employee has worked
At the store for over 10 years?

Accepted Answer

To find the probability that an employee has worked at the store for over 10 years, we use the Z-score formula: Z = (X - μ) / σ, where X is the value of interest (10 years), μ is the mean (5.7 years), and σ is the standard deviation (1.8 years). Plugging in the values gives Z = (10 - 5.7) / 1.8 = 2.39. Looking up 2.39 on a standard normal distribution table or using a calculator gives a right-tail probability of approximately 0.8%.

Question 10

The average gas mileage of a certain model car is 30.0 miles per gallon. If the gas mileages are normally distributed with a standard deviation of 0.75 miles per gallon, find
The probability that a car has a gas mileage of between 29.8 and 30.2 miles per gallon.

Accepted Answer

The answer of The average gas mileage of a certain...

Question 11

In a standard normal distribution, what $z$ value corresponds to $17 \%$ of the data between the mean and the $z$ value?

Accepted Answer

The answer of In a standard normal distribution, what $z$...

Question 12

$$\text { Find the } z \text {-scores that bound the middle } 74 \% \text { of the area under the standard normal curve. }$$

Accepted Answer

The answer of \[\text { Find the } z \text...

Question 13

$$\text { Find the } z \text {-score for which the area to the left is } 0.13 \text {. }$$

Accepted Answer

The z-score corresponding to an area of 0.13 to the left is approximately -1.13, as found in standard normal distribution tables.

Question 14

The average charitable contribution itemized per income tax return in Pennsylvania is $792. Suppose that the distribution of contributions is normal with a standard deviation
Of $103. Find the limits for the middle 50% of contributions.

Accepted Answer

The middle 50% of contributions corresponds to the interquartile range, which is between the 25th percentile (Q1) and the 75th percentile (Q3) in a normal distribution. For a normal distribution, these percentiles are approximately 0.675 standard deviations away from the mean in either direction. Therefore, we calculate the limits as follows: Lower limit = Mean - (0.675 * Standard Deviation) = $792 - (0.675 * $103) ≈ $723 Upper limit = Mean + (0.675 * Standard Deviation) = $792 + (0.675 * $103) ≈ $861

Question 15

A normal population has a mean $\mu = 28$ and standard deviation $\sigma = 5$. What proportion of the population is less than 23 ?

Accepted Answer

To find the proportion of the population less than 23, we calculate the z-score: $ z = \frac{23 - 28}{5} = -1 $. Using the standard normal distribution table, the proportion corresponding to a z-score of -1 is approximately 0.1587.

Question 16

$$\text { Find the } z \text {-score for which the area to the right is } 0.27 \text {. }$$

Accepted Answer

The answer of \[\text { Find the } z \text...

Question 17

The weights of 6-week-old poults (juvenile turkeys) are normally distributed with a mean 8.9 pounds and standard deviation 1.9 pounds. A turkey farmer wants to provide a
Money-back guarantee that her 6-week poults will weigh at least a certain amount. What
Weight should she guarantee so that she will have to give her customer's money back
Only 1% of the time?

Accepted Answer

The answer of The weights of 6-week-old poults (juvenile turkeys)...

Question 18

A bottler of drinking water fills plastic bottles with a mean volume of 998 milliliters (mL) and standard deviation 7 mL. The fill volumes are normally distributed. What
Proportion of bottles have volumes greater than 999 mL?

Accepted Answer

To find the proportion of bottles with volumes greater than 999 mL, we use the Z-score formula: Z = (X - μ) / σ, where X is the value in question (999 mL), μ is the mean (998 mL), and σ is the standard deviation (7 mL). Plugging in the values gives Z = (999 - 998) / 7 = 1 / 7 ≈ 0.143. Looking up this Z-score in a standard normal distribution table or using a calculator, we find that the area to the left of this Z-score is approximately 0.5567, meaning the area to the right (which represents the proportion of bottles with volumes greater than 999 mL) is 1 - 0.5567 = 0.4433, which rounds to 0.4443.

Question 19

$$\text { Find the probability } P ( - 0.72 < z < - 0.11 ) \text { using the standard normal distribution. }$$

Accepted Answer

The answer of \[\text { Find the probability } P...

Question 20

A contractor decided to build homes that will include the middle 80% of the market. If the average size of homes built is 2010 square feet, find the maximum and minimum
Sizes of the homes the contractor should build. Assume that the standard deviation is 90
Square feet and the variable is normally distributed.

Accepted Answer

The answer of A contractor decided to build homes that...

Quiz 6: The Normal Distribution

A bottler of drinking water fills plastic bottles with a mean volume of 1001 milliliters (mL) and standard deviation 6 mL. The fill volumes are normally distributed. What Proportion of bottles have volumes less than 1001 mL?

If a normally distributed group of test scores have a mean of 70 and a standard deviation of 12, find the percentage of scores that will fall below 50.

A normal population has a mean $\mu = 40$ and standard deviation $\sigma = 11$ . What proportion of the population is between 24 and $32 ?$

A normal population has a mean $\mu = 33$ and standard deviation $\sigma = 9$ . What is the probability that a randomly chosen value will be greater than 44 ?

A bottler of drinking water fills plastic bottles with a mean volume of 999 milliliters (mL) and standard deviation 5 mL. The fill volumes are normally distributed. What Proportion of bottles have volumes between 992 mL and 998 mL?

The average hourly wage of workers at a fast food restaurant is $6.50/hr. Assume the wages are normally distributed with a standard deviation of $0.45. If a worker at this fast Food restaurant is selected at random, what is the probability that the worker earns more Than $6.75?

The average length of crocodiles in a swamp is 11.5 feet. If the lengths are normally distributed with a standard deviation of 1.7 feet, find the probability that a crocodile is More than 11 feet long.

At a large department store, the average number of years of employment for a cashier is 5.7 with a standard deviation of 1.8 years, and the distribution is approximately normal. If an employee is picked at random, what is the probability that the employee has worked At the store for over 10 years?

The average gas mileage of a certain model car is 30.0 miles per gallon. If the gas mileages are normally distributed with a standard deviation of 0.75 miles per gallon, find The probability that a car has a gas mileage of between 29.8 and 30.2 miles per gallon.

In a standard normal distribution, what $z$ value corresponds to $17 \%$ of the data between the mean and the $z$ value?

$\text { Find the } z \text {-scores that bound the middle } 74 \% \text { of the area under the standard normal curve. }$

$\text { Find the } z \text {-score for which the area to the left is } 0.13 \text {. }$

The average charitable contribution itemized per income tax return in Pennsylvania is $792. Suppose that the distribution of contributions is normal with a standard deviation Of $103. Find the limits for the middle 50% of contributions.

A normal population has a mean $\mu = 28$ and standard deviation $\sigma = 5$ . What proportion of the population is less than 23 ?

$\text { Find the } z \text {-score for which the area to the right is } 0.27 \text {. }$

A bottler of drinking water fills plastic bottles with a mean volume of 998 milliliters (mL) and standard deviation 7 mL. The fill volumes are normally distributed. What Proportion of bottles have volumes greater than 999 mL?

$\text { Find the probability } P ( - 0.72 < z < - 0.11 ) \text { using the standard normal distribution. }$

Quiz 6: The Normal Distribution

A bottler of drinking water fills plastic bottles with a mean volume of 1001 milliliters (mL) and standard deviation 6 mL. The fill volumes are normally distributed. What Proportion of bottles have volumes less than 1001 mL?

If a normally distributed group of test scores have a mean of 70 and a standard deviation of 12, find the percentage of scores that will fall below 50.

A normal population has a mean μ=40\mu = 40μ=40 and standard deviation σ=11\sigma = 11σ=11 . What proportion of the population is between 24 and 32?32 ?32?

A normal population has a mean μ=33\mu = 33μ=33 and standard deviation σ=9\sigma = 9σ=9 . What is the probability that a randomly chosen value will be greater than 44 ?

A bottler of drinking water fills plastic bottles with a mean volume of 999 milliliters (mL) and standard deviation 5 mL. The fill volumes are normally distributed. What Proportion of bottles have volumes between 992 mL and 998 mL?

The average hourly wage of workers at a fast food restaurant is $6.50/hr. Assume the wages are normally distributed with a standard deviation of $0.45. If a worker at this fast Food restaurant is selected at random, what is the probability that the worker earns more Than $6.75?

The average length of crocodiles in a swamp is 11.5 feet. If the lengths are normally distributed with a standard deviation of 1.7 feet, find the probability that a crocodile is More than 11 feet long.

At a large department store, the average number of years of employment for a cashier is 5.7 with a standard deviation of 1.8 years, and the distribution is approximately normal. If an employee is picked at random, what is the probability that the employee has worked At the store for over 10 years?

The average gas mileage of a certain model car is 30.0 miles per gallon. If the gas mileages are normally distributed with a standard deviation of 0.75 miles per gallon, find The probability that a car has a gas mileage of between 29.8 and 30.2 miles per gallon.

In a standard normal distribution, what zzz value corresponds to 17%17 \%17% of the data between the mean and the zzz value?

Find the z-scores that bound the middle 74% of the area under the standard normal curve. \text { Find the } z \text {-scores that bound the middle } 74 \% \text { of the area under the standard normal curve. } Find the z-scores that bound the middle 74% of the area under the standard normal curve.

Find the z-score for which the area to the left is 0.13. \text { Find the } z \text {-score for which the area to the left is } 0.13 \text {. } Find the z-score for which the area to the left is 0.13.

The average charitable contribution itemized per income tax return in Pennsylvania is $792. Suppose that the distribution of contributions is normal with a standard deviation Of $103. Find the limits for the middle 50% of contributions.

A normal population has a mean μ=28\mu = 28μ=28 and standard deviation σ=5\sigma = 5σ=5 . What proportion of the population is less than 23 ?

Find the z-score for which the area to the right is 0.27. \text { Find the } z \text {-score for which the area to the right is } 0.27 \text {. } Find the z-score for which the area to the right is 0.27.

A bottler of drinking water fills plastic bottles with a mean volume of 998 milliliters (mL) and standard deviation 7 mL. The fill volumes are normally distributed. What Proportion of bottles have volumes greater than 999 mL?

Find the probability P(−0.72<z<−0.11) using the standard normal distribution. \text { Find the probability } P ( - 0.72 < z < - 0.11 ) \text { using the standard normal distribution. } Find the probability P(−0.72<z<−0.11) using the standard normal distribution.

A normal population has a mean $\mu = 40$ and standard deviation $\sigma = 11$ . What proportion of the population is between 24 and $32 ?$

A normal population has a mean $\mu = 33$ and standard deviation $\sigma = 9$ . What is the probability that a randomly chosen value will be greater than 44 ?

In a standard normal distribution, what $z$ value corresponds to $17 \%$ of the data between the mean and the $z$ value?

$\text { Find the } z \text {-scores that bound the middle } 74 \% \text { of the area under the standard normal curve. }$

$\text { Find the } z \text {-score for which the area to the left is } 0.13 \text {. }$

A normal population has a mean $\mu = 28$ and standard deviation $\sigma = 5$ . What proportion of the population is less than 23 ?

$\text { Find the } z \text {-score for which the area to the right is } 0.27 \text {. }$

$\text { Find the probability } P ( - 0.72 < z < - 0.11 ) \text { using the standard normal distribution. }$