## Summarizing Frequency Distributions Exercise

State whether each of the following research examples is best summarized as grouped or ungrouped data. Explain your answer.

1. The time (in seconds) it takes 100 children to complete a cognitive skills game.

2. The number of single mothers with 1, 2, 3, or 4 children.

3. The number of teenagers who have experimented with smoking (yes, no).

4. The age (in years) of freshman students in a local college.

5. The handedness (right- or left-handed) of gifted children.

6. The number of symptoms of stress (ranging between 0 and 12 symptoms) experienced by 180 war veterans.

7. The number of calories in school lunches in a sample of 32 local middle schools.

8. The number of hours of sleep (per night) in a sample of 60 patients being treated for depression.

9. The marital status (single, married, divorced) in a sample of 96 individuals reporting high levels of life satisfaction.

10. The number of traffic violations ticked during a 3-month period in one of four high-crime communities.

State whether a cumulative frequency, relative frequency, relative percent, cumulative relative frequency, or cumulative percent is most appropriate for describing the following situations. For cumulative distributions, indicate whether these should be summarized from the top down or from the bottom up.

11. The frequency of businesses with *at least* 20 employees.

12. The frequency of college students with *less than* a 3.0 GPA.

13. The percentage of women completing 1, 2, 3, or 4 tasks simultaneously.

14. The proportion of pregnancies performed in pubic or private hospitals.

15. The percentage of alcoholics with *more than* 2 years of substance abuse.

16. The proportion of Americans earning $30,000 *at most*.

17. The frequency of hospital visits (per year) in a sample of diabetics.

18. The proportion of elderly patients consuming *at or above* 1,400 calories per day.

19. The percentage of athletes training *at or below* 6 hr per week.

20. The percentage of men in one of six physically demanding occupations. **2.**

## Characteristics of the Mean Exercise

To study perception, a researcher selects a sample of participants (*n* = 12) and asks them to hold pairs of objects differing in weight, but not in size, one in each hand. The researcher asks participants to report when they notice a difference in the weight of the two objects. Below is a list of the difference in weight (in pounds) when participants first noticed a difference. Answer the following questions based on the data given in the table.

Difference in Weight | |

4 | 8 |

9 | 5 |

12 | 7 |

6 | 15 |

10 | 4 |

8 | 8 |

1. State the following values for this set of data:

a) Mean _______

b) Median _______

c) Mode(s) _______

2. What is the shape of this distribution? Hint: Use the values of the mean, median, and mode to infer the shape of this distribution.

3. State whether the mean **increases**, **decreases**, or has **no change** in each of the following scenarios.

a) A 13th value of 10 is added to the data. _______________

b) A 13th value of 4 is added to the data. _______________

c) The value of 8 is removed from the data. _______________

d) A 13th value of 8 is added to the data. _______________

e) The value of 12 is removed from the data. _______________

f) The value of 7 is removed from the data. _______________

4. State the new value of the mean in each of the following scenarios.

a) Each value is multiplied by 4. _______________

b) Each value is divided by 2. _______________

c) 8 is added to each value. _______________

d) 4 is subtracted from each value. _______________

5. Subtract the mean from each score. What is the sum of the differences?

**3.**

## Explaining Sample Variance Exercise

1. What are the degrees of freedom for sample variance in the following samples? Hint: The degrees of freedom are located in the denominator of the sample variance formula.

a) *n* = 50

b) *n* = 10

c) *n* = 12

d) *n* = 120

e) *n* = 75

2.

Suppose we select a sample of size 20 from the following populations. Given the following variances for each population, what will the sample variance be on average when we compute the following formula: ?

a)

= 86

b)

= 121

c)

= 144

d)

= 329

e)

= 600

3.

Suppose we select a sample of size 200 (instead of a sample of size 20) from each population (a to e) listed in Question 2. For each population, what will the sample variance equal, on average, when we compute the following formula: ?

**4.**

## Probability and Conditional Probabilities Exercise

Researchers are often interested in the likelihood of sampling outcomes. They may ask questions about the likelihood that a person with a particular characteristic will be selected to participate in a study. In this exercise, we will select a sample of one participant from the following hypothetical student population of men and women living on or off campus. The population is summarized in the following table.

Male | Female | Row Totals | |

On campus | 30 | 25 | 55 |

Off campus | 20 | 25 | 45 |

Column Totals | 50 | 50 | 100 |

1. What is the probability of selecting a male participant?

2. What is the probability of selecting a female participant?

3. What is the probability of selecting a student who lives on campus?

4. What is the probability of selecting a student who lives off campus?

5. What is the probability of selecting a male student, given that he lives off campus?

6. What is the probability of selecting a female student, given that she lives on campus?

7. What is the probability of selecting a male student, given that he lives on campus?

8. What is the probability of selecting a female student, given that she lives off campus?

9. What is the probability of selecting a student who lives on campus, given that he is a male?

10. What is the probability of selecting a student who lives off campus, given that he is a male?

11. What is the probability of selecting a student who lives on campus, given that she is a female?

12. What is the probability of selecting a student who lives off campus, given that she is a female?

**5.**

## Sampling Distribution of the Mean Exercise

Suppose a researcher wants to learn more about the mean attention span of individuals in some hypothetical population. The researcher cites that the attention span (the time in minutes attending to some task) in this population is normally distributed with the following characteristics: 20 36 . Based on the parameters given in this example, answer the following questions:

1. What is the population mean (μ)?

2.

What is the population variance ?

3. Sketch the distribution of this population. Make sure you draw the shape of the distribution and include the mean plus and minus three standard deviations.

Now say this researcher takes a sample of four individuals (*n* = 4) from this population to test whether the mean attention span in this population is really 20 minutes attending to some task.

4.

What is the mean of the sampling distribution for samples of size 4 from this population? Note: The mean of the sampling distribution is .

5.

What is the standard error for this sampling distribution? Note: The standard error of the sampling distribution is .

6. Based on your calculations for the mean and standard error, sketch the sampling distribution of the mean taken from this population. Make sure you draw the shape of the distribution and include the mean plus and minus three standard errors.

7. If a researcher takes one sample of size 4 (*n* = 4) from this population, what is the probability that he or she computes a sample mean of at least 23 (*M* = 23) minutes? Note: You must compute the *z*-transformation for sampling distributions, and then refer to the unit normal table to find the answer.

8. Now say a researcher decides to take a sample of size 36 (*n* = 36) and calculates the same sample mean (*M* = 23).

a. Compute the mean and standard error for this new sampling distribution. Sketch the sampling distribution taken from this same population. Make sure you draw the shape of the distribution and include the mean plus and minus three standard errors. Hint: A larger sample size will decrease the standard error of the mean.

b. Now what is the probability of obtaining at least this sample mean of 23 minutes when *n* = 36?

9. From which sample (*n* = 4 or *n* = 36) is there a lower probability of collecting a sample mean of 23 from a population with a mean of 20?

10. What does your answer to question 9 indicate about the relationship between sample size and standard error?

**6.**

## Power Exercise

1. Two researchers make a test concerning the effectiveness of a drug use treatment. Researcher A determines that the effect size in the population of males is *d* = 0.36; Researcher B determines that the effect size in the population of females is *d* = 0.20. All other things being equal, which researcher has more power to detect an effect? Explain.

2. Two researchers make a test concerning the levels of marital satisfaction among military families. Researcher A collects a sample of 22 married couples (*n* = 22); Researcher B collects a sample of 40 married couples (*n* = 40). All other things being equal, which researcher has more power to detect an effect? Explain.

3.

Two researchers make a test concerning standardized exam performance among senior high school students in one of two local communities. Researcher A tests performance from the population in the northern community, where the standard deviation of test scores is 110 (); Researcher B tests performance from the population in the southern community, where the standard deviation of test scores is 60 (). All other things being equal, which researcher has more power to detect an effect? Explain.

4. Explain how your answer to question 3 is related to standard error. Hint: You need to refer to the formula for computing standard error in order to answer this question.

5.

Two researchers make a test concerning the same hypothesis. Researcher A tests her hypothesis at a .05 level of significance (); Researcher B tests his hypothesis at a .01 level of significance (). All other things being equal, which researcher has more power to detect an effect? Explain.

6.

Two researchers make a test concerning the effects of music on mood. Researcher A tests the effects of rock music on mood and determines that the likelihood of committing a Type II error is = .20; Researcher B tests the effects of classical music on mood and determines that the likelihood of committing a Type II error is = .40. All other things being equal, which researcher has more power to detect an effect? Explain.

**7.**

## The *t* Table Exercise

Work through the following exercise and write down what you see in the *t*-table. This will help familiarize you with the table.

The *t*-table: The degrees of freedom (*df*) are listed down the far left column. The level of significance for a one-tailed test (top) and two-tailed test (bottom) is listed for each column to the right of the degrees of freedom. So to read the table, you simply need to know the level of significance and degrees of freedom (*n* –1) for the test.

## Increasing the level of significance and degrees of freedom

1. For the following degrees of freedom, list the critical values for a two-tailed test at a .01, .05, and .10 level of significance.

.01 .05 .10

*df* = 8 ___ ___ ___

*df* = 20 ___ ___ ___

*df* = 40 ___ ___ ___

*df* = 120 ___ ___ ___

2. As the level of significance increases (from .01 to .10), does the critical value increase or decrease?

3. As the degrees of freedom increase (from 8 to 120), does the critical value increase or decrease?

4. How is this related to power? Hint: If you are having trouble, refer to Sections 8.8 and 8.9 in Chapter 8 for a review of factors that influence power.

## Comparing the *z* and the *t*

5. For the following degrees of freedom, list the critical values for a one-tailed and two-tailed test at a .01 and .05 level of significance.

.01 .05

*df* = ___ ___ (One-tailed test)

*df* = ___ ___ (Two-tailed test)

6. Refer to Table 8.4 in Chapter 8. How do the critical values you listed in the previous question compare with those listed for the *z*-distribution. Explain.

**8.**

## Independent or Dependent Samples Exercise

For each of the following research questions, you will read an independent and dependent samples study to test that research question. Identify which study used independent and which used dependent samples.

1. How does winning influence self-efficacy (or belief in one’s ability) among high school sports athletes?

Study 1: Researchers hypothesized that the more wins a team has, the more teammates will believe in their own abilities (self-efficacy). To test this, they selected a group of sports athletes from teams with few wins during the previous season and a second group of sports athletes from teams with many wins during the previous season. They tested whether self-efficacy significantly differed between groups.

Study 2: Researchers hypothesized that the more wins a team has, the more teammates will believe in their own abilities (self-efficacy). To test this, they measured self-efficacy among athletes during a winning season and again during a losing season the following year. They tested whether the difference in self-efficacy between winning and losing seasons was significantly different.

2. Will levels of the brain chemical dopamine increase in response to food rewards?

Study 1: Researchers hypothesized that dopamine would increase in response to a high-fat food (reward) versus a similar low-fat food. To test this, they presented four adult female *Macaca fascicularis* monkeys with a high- and low-fat raisin and measured the levels of dopamine in the brain in response to each reward. They tested whether the difference in dopamine levels was significantly different.

Study 2: Researchers hypothesized that the levels of dopamine would increase in response to a larger food reward. To test this, they had animal subjects press a lever to receive a food reward and measured levels of dopamine in the brain in response to this reward. One group received a small food reward and a second group received a large food reward. They tested whether dopamine levels significantly differed between groups.

3. Do English as a second language (ESL) children have the same knowledge of word meanings as English as a first language (ELI) children?

Study 1: Researchers hypothesized that ESL children would have greater difficulty in phonological awareness (the ability to notice, think about, or manipulate the individual sounds in words). To test this, they had one group of ESL children and a second group of ELI children complete a phonological awareness assessment and recorded the scores. They tested whether phonological awareness assessment scores significantly differed between groups.

Study 2: Researchers hypothesized that ESL children would have the same knowledge of word meanings as do their ELI relatives. To test this, they recorded scores on a vocabulary knowledge assessment for ESL children and comparison ELI relatives (of similar ages). They tested whether the difference in vocabulary knowledge assessment scores between these related children was significantly different.

4. How do psychosocial attitudes (those based on the interaction between the individual and the social environment) toward romantic love vary by race/culture?

Study 1: Researchers hypothesized that cultures focused on group identity (collectivist) would place less emphasis on romantic love compared with cultures focused on individual identity (individualistic). To test this, they collected a sample of Chinese (collectivist) and American (individualistic) adolescents and had them complete a romantic love assessment survey. They tested whether scores on this survey significantly differed between groups.

Study 2: Researchers hypothesized that psychosocial attitudes toward love in individualistic cultures would be more negative following the loss of a romantic relationship. To test this, they recorded perceptions of love among a group of American college students during and following a romantic relationship. They tested whether the difference in perceptions of love during and after a romantic relationship was significantly different.

5. Does snacking between meals reduce the number of calories consumed in a meal?

Study 1: Researchers hypothesized that snacking between meals would only reduce calories consumed in the meal if participants thought of the foods as “snack foods” (referred to as cognitive satiety). To test this, on 1 day, they gave a group of participants 100 kcal of a food they thought of as a snack food and recorded the number of calories consumed in the subsequent meal. On the second day, they gave the same group of participants 100 kcal of a food they thought of as a “meal food” and recorded the number of calories consumed in the subsequent meal. They tested whether the difference in calories consumed in the meal from day 1 to day 2 significantly differed.