Question 1

Data Set 7-4: A set of scores with a mean of 16 and a standard deviation of 3.
-Look at Data Set 7-4. What scores are so extreme they are made by only five percent of the population?

Accepted Answer

Scores that are in the extreme 5% of a normal distribution are typically found beyond 1.645 standard deviations from the mean in either direction. Calculating this: 16 ± (1.645 * 3) gives approximately 10.12 and 21.88.

Question 2

Data Set 7-4: A set of scores with a mean of 16 and a standard deviation of 3.
-Look at Data Set 7-4. What proportion of the population has scores between 12 and 17, including both 12 and 17?

Accepted Answer

The answer of Data Set 7-4: A set of scores...

Question 3

Data Set 7-4: A set of scores with a mean of 16 and a standard deviation of 3.
-Look at Data Set 7-4. Suppose 200 persons received scores during a period of a week. How many would have scores of 15 or larger?

Accepted Answer

Given the mean is 16 and the standard deviation is 3, a score of 15 is one-third of a standard deviation below the mean. In a normal distribution, about 50% of the values lie above the mean, and since 15 is very close to the mean, slightly more than 50% of the 200 persons would have scores of 15 or larger. None of the provided numerical options accurately reflect this, making D the correct choice.

Question 4

Data Set 7-5: Suppose you knew that a population had a mean of 150 and a standard deviation of 50.
-Look at Data Set 7-5. To use Table C in the back of your text, you must assume that

Accepted Answer

To use Table C (typically a z-table or standard normal distribution table), the assumption that the population is normally distributed is necessary. This allows for the application of the empirical rule and z-score calculations, which are based on the normal distribution properties.

Question 5

Data Set 7-5: Suppose you knew that a population had a mean of 150 and a standard deviation of 50.
-Use Table C to answer questions about Data Set 7-5. The proportion of the population with scores less than 225 is

Accepted Answer

To find the proportion of the population with scores less than 225, you calculate the z-score using the formula z = (X - μ) / σ, where X is the value in question, μ is the mean, and σ is the standard deviation. For X = 225, μ = 150, and σ = 50, the z-score is (225 - 150) / 50 = 1.5. Looking up a z-score of 1.5 in Table C (a standard normal distribution table), you find the proportion of the population below this z-score is approximately .9332.

Question 6

Data Set 7-5: Suppose you knew that a population had a mean of 150 and a standard deviation of 50.
-Use Table C and Data Set 7-5. The proportion of the population with scores less than 0 is

Accepted Answer

To find the proportion of the population with scores less than 0, we need to calculate the z-score for 0 using the formula z = (X - μ) / σ, where X is the score, μ is the mean, and σ is the standard deviation. Here, z = (0 - 150) / 50 = -3. Using the standard normal distribution table (Table C), a z-score of -3 corresponds to a proportion of 0.0013.

Question 7

Data Set 7-5: Suppose you knew that a population had a mean of 150 and a standard deviation of 50.
-If Data Set 7-5 were based on 500 scores, the number with scores between 175 and 200 would be

Accepted Answer

To find the number of scores between 175 and 200, we can use the standard normal distribution. First, we convert the scores to z-scores: $z = \frac{X - \mu}{\sigma}$, where $X$ is the score, $\mu$ is the mean, and $\sigma$ is the standard deviation. For 175, $z = \frac{175 - 150}{50} = 0.5$, and for 200, $z = \frac{200 - 150}{50} = 1$. Looking at a standard normal distribution table, the area between $z = 0.5$ and $z = 1$ is approximately 0.2122, representing the proportion of the population between these scores. Multiplying this by the total number of scores (500) gives $500 \times 0.2122 = 106$, indicating that approximately 106 scores are between 175 and 200.

Question 8

Data Set 7-5: Suppose you knew that a population had a mean of 150 and a standard deviation of 50.
-The proportion of the population of scores in Data Set 7-5 that is between 100 and 175 would be

Accepted Answer

To find the proportion of the population with scores between 100 and 175, we use the standard deviation and mean to convert these scores into z-scores. For 100: $z = \frac{100 - 150}{50} = -1$, and for 175: $z = \frac{175 - 150}{50} = 0.5$. Looking up these z-scores in a standard normal distribution table, the area between z = -1 and z = 0 is 0.3413, and the area between z = 0 and z = 0.5 is 0.1915. Adding these areas together gives 0.3413 + 0.1915 = 0.5328. However, the closest option provided is D) .4706, which suggests a calculation or interpretation error in the explanation. The correct approach involves finding the area under the standard normal curve between the z-scores of -1 and 0.5. The correct calculation directly finds this area or closely approximates it, which is not accurately reflected in the provided options. The explanation aimed to demonstrate the process of converting raw scores to z-scores and summing areas under the normal curve, but the precise answer should align with standard normal distribution properties, indicating a need for correction in the explanation or reevaluation of the options based on accurate z-score areas.

Question 9

Data Set 7-6, a theoretical distribution: 
-Look at Data Set 7-6. The probability of occurrence of Event B or Event E is

Accepted Answer

The answer of Data Set 7-6, a theoretical distribution: 
-Look...

Question 10

Data Set 7-6, a theoretical distribution: 
-Look at Data Set 7-6. This distribution is

Accepted Answer

The answer of Data Set 7-6, a theoretical distribution: 
-Look...

Question 11

Data Set 7-6, a theoretical distribution: 
-Look at Data Set 7-6. The probability of occurrence of Events B or C is

Accepted Answer

The answer of Data Set 7-6, a theoretical distribution: 
-Look...

Question 12

The difference between an empirical distribution and a theoretical distribution is that a theoretical distribution

Accepted Answer

A theoretical distribution is based on mathematical models and logic, describing how outcomes are expected to occur under certain conditions, rather than being directly observed from empirical data.

Question 13

A characteristic that always distinguishes an empirical from a theoretical curve is

Accepted Answer

Empirical curves are based on observed data, while theoretical curves are based on a hypothesis or theory. The method of obtaining the points (observed data vs. theoretical predictions) is what distinguishes an empirical curve from a theoretical one.

Question 14

An empirical distribution is based on

Accepted Answer

An empirical distribution is based on observations, as it is derived from actual data collected through experiments or real-world collection methods.

Question 15

An empirical distribution is based on

Accepted Answer

An empirical distribution is based on observation, as it is derived from observed data rather than theoretical models or formulas.

Question 16

Which of the following is a theoretical distribution?

Accepted Answer

A theoretical distribution is one that is based on a theoretical model rather than actual data. Choice C, a frequency polygon of the ages of all students in SHES four years from now, is theoretical because it predicts future data based on assumptions or models, rather than presenting data that has already been collected. Choices A, B, and D are based on actual data collected from past or current events.

Question 17

Which of the following is a theoretical distribution?

Accepted Answer

Theoretical distributions are abstract concepts that represent the probabilities of various outcomes in a perfect scenario, rather than counts of actual events or items like cups of coffee consumed over time.

Question 18

Which of the following is a theoretical distribution?

Accepted Answer

Theoretical distributions are not based on actual data collection or empirical evidence but on mathematical models or assumptions. Choices A, B, and C all involve collecting or observing real-world data, which makes them empirical rather than theoretical distributions.

Question 19

Which of the following is an empirical distribution?

Accepted Answer

An empirical distribution is based on observed data rather than theoretical calculations. The given names and their frequencies of all high school graduates in the United States for the year 2010 is an example of observed data, making it an empirical distribution. Choices B and C refer to theoretical distributions.

Question 20

Suppose that should k occur, it will be called a success. If j should occur, it will be called a failure. The ratio of k/(k + j) is

Accepted Answer

The answer of Suppose that should k occur, it will...

Quiz 7: Theoretical Distributions Including the Normal Distribution

Data Set 7-4: A set of scores with a mean of 16 and a standard deviation of 3. -Look at Data Set 7-4. What scores are so extreme they are made by only five percent of the population?

Data Set 7-4: A set of scores with a mean of 16 and a standard deviation of 3. -Look at Data Set 7-4. What proportion of the population has scores between 12 and 17, including both 12 and 17?

Data Set 7-4: A set of scores with a mean of 16 and a standard deviation of 3. -Look at Data Set 7-4. Suppose 200 persons received scores during a period of a week. How many would have scores of 15 or larger?

Data Set 7-5: Suppose you knew that a population had a mean of 150 and a standard deviation of 50. -Look at Data Set 7-5. To use Table C in the back of your text, you must assume that

Data Set 7-5: Suppose you knew that a population had a mean of 150 and a standard deviation of 50. -Use Table C to answer questions about Data Set 7-5. The proportion of the population with scores less than 225 is

Data Set 7-5: Suppose you knew that a population had a mean of 150 and a standard deviation of 50. -Use Table C and Data Set 7-5. The proportion of the population with scores less than 0 is

Data Set 7-5: Suppose you knew that a population had a mean of 150 and a standard deviation of 50. -If Data Set 7-5 were based on 500 scores, the number with scores between 175 and 200 would be

Data Set 7-5: Suppose you knew that a population had a mean of 150 and a standard deviation of 50. -The proportion of the population of scores in Data Set 7-5 that is between 100 and 175 would be

Data Set 7-6, a theoretical distribution: -Look at Data Set 7-6. The probability of occurrence of Event B or Event E is

Data Set 7-6, a theoretical distribution: -Look at Data Set 7-6. This distribution is

Data Set 7-6, a theoretical distribution: -Look at Data Set 7-6. The probability of occurrence of Events B or C is

The difference between an empirical distribution and a theoretical distribution is that a theoretical distribution

A characteristic that always distinguishes an empirical from a theoretical curve is

An empirical distribution is based on

An empirical distribution is based on

Which of the following is a theoretical distribution?

Which of the following is a theoretical distribution?

Which of the following is a theoretical distribution?

Which of the following is an empirical distribution?

Suppose that should k occur, it will be called a success. If j should occur, it will be called a failure. The ratio of k/(k + j) is