Question 1

The weight of potato chip bags filled by a machine at a packaging plant is normally distributed, with a mean of 15.0 ounces and a standard deviation of 0.2 ounces. What is the probability that a randomly chosen bag will weigh less than 14.6 ounces?

Accepted Answer

To find the probability that a bag weighs less than 14.6 ounces, we calculate the z-score for 14.6 ounces using the formula z = (X - μ) / σ, where X is the value (14.6 ounces), μ is the mean (15.0 ounces), and σ is the standard deviation (0.2 ounces). This gives us z = (14.6 - 15.0) / 0.2 = -2. The z-score of -2 corresponds to a probability of about 2.5% in the standard normal distribution table.

Question 2

Which of the following is true about any normal distribution?

Accepted Answer

A normal distribution is characterized by its symmetrical and bell-shaped curve, which means that the majority of the data falls near the mean, with fewer and fewer data points spread out toward the tails. There is no restriction on negative values, and the standard deviation can be any value. While the mean is an important characteristic of a normal distribution, it is not the only defining feature.

Question 3

The shelf life of a battery produced by one major company is known to be normally distributed, with a mean life of 3.5 years and a standard deviation of 0.75 years. What percent of batteries last between 1.25 and 5.75 years?

Accepted Answer

The range from 1.25 to 5.75 years is 3 standard deviations below and above the mean (3.5 years) in a normal distribution, which encompasses approximately 99.7% of the data according to the empirical rule.

Question 4

The mean length of time, per week, that students at a certain school spend on their homework is 24.3 hours, with a standard deviation of 1.4 hours. Assuming the distribution of study times is normal, what percent of students study between 22.9 and 25.7 hours?

Accepted Answer

This question involves the empirical rule (68-95-99.7 rule) for normal distributions, which states that approximately 68% of the data falls within one standard deviation of the mean. Here, 22.9 hours and 25.7 hours are one standard deviation away from the mean (24.3 hours ± 1.4 hours), so approximately 68% of students study between these times.

Question 5

The length of students' college careers at Anytown University is known to be normally distributed, with a mean length of 5.5 years and a standard deviation of 1.75 years. What percent of students have college careers lasting between 2 and 9 years?

Accepted Answer

We need to find the percentage of students whose college careers lasted between 2 and 9 years. Using the z-score formula, we can find the z-scores for both 2 and 9 years:
z-score for 2 years: (2-5.5)/1.75 = -2
z-score for 9 years: (9-5.5)/1.75 = 2
From the z-tables, we can find that the area under the standard normal distribution curve between -2 and 2 is approximately 0.95, or 95%. Therefore, approximately 95% of students have college careers lasting between 2 and 9 years. The correct choice is D.

Question 6

The weight of potato chip bags filled by a machine at a packaging plant is normally distributed, with a mean of 15.0 ounces and a standard deviation of 0.2 ounces. What is the probability that a randomly chosen bag will weigh more than 15.6 ounces?
A)5%
B)2.5%
C)0.3%
D)0.15%

Accepted Answer

The answer of The weight of potato chip bags filled...

Question 7

For the following Questions assume that the heights of American women ages 18 to 24 are normally distributed, with mean 64.5 inches and standard deviation 2.5 inches.
-Twenty-year-old Amy is 67 inches tall. Where is she located in the height distribution of women ages 18 to 24?

Accepted Answer

Amy's height is 67 inches, which is above the mean of 64.5 inches. Given the normal distribution and a standard deviation of 2.5 inches, a height of 67 inches is more than one standard deviation above the mean but less than two standard deviations above. In a normal distribution, the first quartile is below one standard deviation below the mean, and the third quartile is below one standard deviation above the mean. Therefore, Amy's height places her above the third quartile.

Question 8

For the following Questions assume that the heights of American women ages 18 to 24 are normally distributed, with mean 64.5 inches and standard deviation 2.5 inches.
-Amy's 18-year-old sister Mo is 64 inches tall. Where is she located in the height distribution of women ages 18 to 24?

Accepted Answer

Since Mo is shorter than the mean height (64.5 inches), she is located between the first quartile (Q1) and the mean.

Question 9

For the following Questions assume that the heights of American women ages 18 to 24 are normally distributed, with mean 64.5 inches and standard deviation 2.5 inches.
-About what percentage of American women ages 18 to 24 have heights greater than 69.5 inches?

Accepted Answer

To find the percentage of women taller than 69.5 inches, we calculate the z-score: $z = \frac{69.5 - 64.5}{2.5} = 2$. Looking up a z-score of 2 in a standard normal distribution table, we find that about 97.5% of the area is to the left, meaning 2.5% is to the right, or taller than 69.5 inches.

Question 10

Bruce is considering relocating to a new city. His options are Altuna, Boundary Valley, and Collie Creek. All other living expenses seem to be equal for the three choices, but Bruce is worried about his heating and cooling bill. The only information he has at his disposal is the following table: $\begin{array} { l c c } 
\text { City } & \begin{array} { c } 
\text { Mean daily } \
\text { temperature }
\end{array} & \begin{array} { c } 
\text { Standard deviation for } \
\text { the daily temperature }
\end{array} \
\text { Altuna } & 64 & 15 \
\text { Boundary Valley } & 64 & 16 \
\text { Collie Creek } & 64 & 5
\end{array}$ Which city is likely to have the least expensive heating and cooling costs?

Accepted Answer

The standard deviation for the daily temperature is a measure of how much variation there is in the temperature from day to day. The lower the standard deviation, the more consistent the temperature is. Collie Creek has the lowest standard deviation, indicating that the temperature is more consistent from day to day. This means that there will be fewer extreme temperature days that require more heating or cooling, resulting in lower costs.

Question 11

A data set consists of integer values. If the value 7 has relative frequency 2/11, which statement must be true?

Accepted Answer

The relative frequency of a value is the number of times that value appears divided by the total number of observations in the data set. If the relative frequency of 7 is 2/11, it means that 7 appears $2/11 \times \text{total number of observations}$ times. Since you can't have a fraction of an occurrence, this means at least 2 occurrences of 7 are in the data set, making option C correct. Options A and B make specific claims about the total number of values in the data set, which cannot be determined from the given relative frequency alone. Option D assumes 7 is the mode, which is not necessarily true without knowing the frequency of other values in the data set.

Question 12

The annual income of residents in a certain county is normally distributed, with a mean of $42,000 and a standard deviation of $10,000. What is the probability that a randomly chosen resident has income over $52,000?

Accepted Answer

To find the probability of a resident earning over $52,000, we calculate the Z-score: Z = (X - μ) / σ = (52,000 - 42,000) / 10,000 = 1. A Z-score of 1 corresponds to the top 16% of the normal distribution, hence the probability is 16%.

Question 13

The distribution of home prices in Nirvana is skewed to the left. The median price is $167,900. The mean is:

Accepted Answer

The distribution is skewed to the left, which means that there are some lower-priced homes that are pulling down the mean. Therefore, the mean is lower than the median price of $167,900.

Question 14

If the values in a data set are all multiplied by -1, what happens to the standard deviation?

Accepted Answer

The standard deviation is a measure of the spread of the data. If all the values in a data set are multiplied by the same constant, the spread of the data will not change. Therefore, the standard deviation will remain unchanged.

Question 15

If the values in a data set are all multiplied by 3, what happens to the standard deviation?

Accepted Answer

When all the values in a data set are multiplied by a constant, the standard deviation is also multiplied by the same constant. Therefore, if all the values are multiplied by 3, the standard deviation will also be multiplied by 3.

Quiz 5: Exploring Data: Distributions