Assignment 1 Descriptive Statistics (due 2/4/15-2/9/15) (13. 5 pts)

Computation by hand

3. 
   Compute the sample standard deviation (3).
   

d. How many students scored at 17? (.5)

e. How many students scored below 12? (.5)

f. What percentage of people scored 5 or below 5? (.5)

1. 
   What is the mean? (.5)
   

2. 
   What is the standard deviation? Is the standard deviation SPSS computed the same as the one you computed by hand (except rounding error)? (.5)
   

3. 
   Fifty percent of students performed better than what score? (.5)
   

4. 
   What is the mode? (.5)
   

5. 
   Using SPSS, create a histogram based on these test scores. Impose the normal curve over your histogram. Locate the mean, mode, and median you computed in Question 1 in the histogram by hand. Does the score distribution for Instructor Jill’s class approximate the normal distribution? If not, what is the shape of the distribution? (2)
   

6. 
   Based on these analyses, what you think Instructor Jill should be thinking about her students’ performance? If you were Jill, what reasons(s) do you give to the performance of her students? (1)
   

1. 
   Please see Appendix D (p. 737-739) for data entry.
   
2. 
   If you want to name your variable (instead of using var00001), click on the Variable View tab at the bottom of the data editor. Type in a new name (for example, score) in the name box. Click the Data View tab to go back to the data editor. Did the variable name change?
   
3. 
   Click Analyze on the tool bar, select Descriptive Statistics, and click on Frequencies.
   
4. 
   Move your variable in the left box into the Variable box (right box) by clicking the arrow mark in the middle.
   
5. 
   Click Statistics, select Mean, Median, Mode, and Std. deviation, and click Continue.
   
6. 
   Click Charts, select Histogram, and with normal curve, and click Continue.
   

Your name _________________________

b. Compute μ and σ for the population and locate them in the above graph (3).

c. Transform each of the scores above into a z-score (3).

d. Draw a frequency distribution graph of the z-scores you computed. Did the shape of the distribution of the z-score change from the original shape of the distribution of original scores? (1)

e. Compute the mean and standard deviation of these z-scores by hand (2.5).

2. 
   Describe exactly what a z-score measures and what information it provides (1).
   

3. 
   A distribution of exam scores has a mean of μ= 80.
   

1. 
   If your score is X = 86, which standard deviation would give you a better grade: σ = 4 or σ = 8? Explain your answer (1).
   

2. 
   If your score is X = 74, which standard deviation would give you a better grade: σ = 4 or σ = 8? Explain your answer (1).
   

4. 
   Assume that there is an animal intelligence test. A psychologist administers it to a representative sample of cats and a representative sample of dogs. Both species turn out to be, on average, equally smart. (That is, M_dogs = M_cats). But, more variability occurred in the IQs of dogs than cats. (That is s_dogs > s_cats). If we consider animals with IQs about 70 to be “geniuses,” will there be more genius cats or more genius dogs? Explain your answer. (Hint: If you draw the distribution of IQ score for dogs and cats, you can see the answer) (1).
   

4. 
   A local hardware store has a “Savings Wheel” at the checkout. Customers get to spin the wheel and, when the wheel stops, a pointer indicates how much they will save. The wheel can stop in any one of 50 sections. Of the sections, 10 produce 0% off, 20 sections are for 10% off, 10 sections are for 20% off, 5 for 30% off, 3 for 40% off, 1 for 50%, and 1 for 100%. Assuming that all 50 sections are equally likely,
   

1. 
   What is the probability that a customer’s purchase will be free (100%)?
   

2. 
   What is the probability that a customer will get no savings from the wheel (0% off)? _______________ (.5)
   

2. 
   What is the probability that a customer will get at least 20% off?
   

5. 
   A psychology class consists of 14 males and 36 females. If the professor selects names from the class listing using random sampling, (1)
   

1. 
   What is the probability that the first student selected will be a female? ___________
   

2. 
   If a random sample of n = 3 students (sequential selection) is selected and the first two are both females, what will be the probability that the third child selected will be male? ___________
   

5. 
   For a normal distribution, identify the nearest z-score that would separate the
   

7. A recent newspaper article reported the results of a survey of well-educated suburban parents. The responses to one question indicated that by age 2, children were watching an average of μ = 60 minutes of television each day. Assuming that the distribution of television-watching times is normal with a standard deviation of σ = 20 minutes, find each of the following proportions.

b. What proportion of 2-year-old children watch less than 20 minutes a day? Is it a likely event? (1.5)

Your name ___________________________

b. What is the expected value of M and the standard error for a sample of n = 100 individuals? (1)

c. What value (the expected value of M or the standard error) did change if you changed the sample size? (1)

2. A distribution of scores has  = 15, but the value of the population mean is unknown. You plan to select a sample from the population in order to learn more about the unknown population mean.

1. 
   If your sample consists of n = 25 scores, how much error, on average, would you
   

b. If your sample consists of n = 100, how much error, on average, would you expect

between your sample mean and the population mean (1)?

c. Would you prefer a sample of n = 25 or a sample of n = 100 to estimate the

unknown population mean accurately?  Explain your answer. (1)

4. The population of IQ scores forms a normal distribution with μ = 100 and σ = 15.

1. 
   What is the probability of obtaining a sample mean greater than M = 105 for a random sample of n = 9 scores? Is it a likely event? (Hint: Sketch the distribution of sample means, locate he expected value of M and the standard error in your sketch, locate the sample mean and shade the area you want) (1.5).
   

b. What is the probability of obtaining a sample mean greater than M = 105 for a random sample of n = 36 scores? Is it a likely event? (1.5)

3. 
   Why is the probability smaller when you have a random sample of n = 36 than when you have a random sample of n = 9? (1)
   

5. 
   People are selected to serve on juries by randomly picking names from the list of registered voters. The average age for registered voters in the county is μ = 44.3 years with a standard deviation of σ = 12.4. A statistician computes the average age for a group of n = 12 people currently serving on a jury and obtains a mean of M = 48.9 years.
   

1. 
   How likely is it to obtain a random sample of n = 12 jurors with an average age equal to or greater than 48.9? (1.5)
   

2. 
   Is it reasonable to conclude that this set of n = 12 people is not a representative random sample of registered voters? Explain your answer (1).
   

Your name ______________________________

d. State your statistical decision. (.5)

e. What is your conclusion? Describe in words and interpret the result (don't forget the statistical information) (2).

f. Given your statistical decision (in part d), what type of decision error could you have made and what is the probability of making that error? (.5)

g. Compute Cohen’s d to measure the size of the effect. Interpret what this effect size really means in this context (don’t just say “large effect” or “small effect”). (2)

h. If you changed the alpha level from .05 to .01, would that affect your statistical decision? Explain your answer. (.5)
i. If the alpha level is changed from .05 to .01, what happens to the boundaries for the critical region? (.5)

j. If the alpha level is changed from .05 to .01, what happens to the probability of a Type I error? (.5)

2. 
   A researcher is testing the hypothesis that consuming a sports drink during exercise will improve endurance. A sample of n = 49 male college students is obtained and each student is given a series of three endurance tasks and asked to consume 4 ounces of the drink during each break between tasks. The overall endurance score for this sample is M = 54. For the general population, without any sports drink, the scores for this task average µ = 50 with a standard deviation of σ = 12. Can the researcher conclude that endurance scores with the sports drink are significantly higher than scores without the drink? Use a one-tailed test with α = .05.
   

a. State the null hypothesis in words and in a statistical form. (1)

c. Compute the appropriate statistic to test the hypotheses. Sketch the distribution with the standard error and locate the critical region with the critical value (3).

d. State your statistical decision. (.5)

5. 
   Compute Cohen’s d to measure the size of the effect. Interpret what this effect size
   

5. 
   What is your conclusion? Describe in words (don't forget the statistical information) (2).
   

2. 
   a. A journalist was comparing the horsepower of a sample of contemporary American cars to the population value of horsepower for American cars of the 1970s. He concluded that there was a statistically significant difference, such that contemporary cars had more horsepower. Unfortunately, his conclusion was error. What type of error did the journalist commit? __________________________ (.5)
   

2. 
   A Verizon researcher compared the number of text messages sent by a sample of teenage boys to the population mean for all Verizon users. She found no evidence to conclude that there was a difference. Unfortunately, her conclusion was in error. What kind of error did she make? _____________________________ (.5)
   

Your name _____________________________

1. 
   A psychologist would like to determine whether there is a relationship between depression and aging. It is known that the general population averages μ = 40 on a standardized depression test. The psychologist obtains a sample of n = 9 individuals who are all more than 70 years old. The depression scores for this sample are as follows: 37, 50, 43, 41, 39, 45, 49, 44, and 48. On the basis of this sample, is depression for elderly people significantly different from depression in the general population? Use a two-tailed test α = .05.
   

1. 
   State the null hypothesis in words and in a statistical form (1).
   

2. 
   State the alternative hypothesis in words and a statistical form (1).
   

3. 
   Compute the appropriate statistic to test the hypotheses. Sketch the distribution with the estimated standard error and locate the critical region(s) with the critical value(s) (6).
   

4. 
   State your statistical decision. (.5)
   

5. 
   Compute Cohen’s d. Interpret what this d really means in this context. (2)
   

f. Compute 95% CI (2).

g. What is your conclusion? Describe in words and in a statistical form (e.g., t-score, df, type of test, α, Cohen’s d) and interpret the result. (2)

2. 
   An educational psychologist was interested in time management by students. She had a theory that students who did well in school spent less time involved with online social media. She found out, from the American Social Media Research Collective, that the average American high school student spends μ = 18.68 hours per week using online social media. She then obtained a sample of n = 31 students who had been named the valedictorians of their high schools. These valedictorians spent an average of M = 16.24 hours using online social media every week with the standard deviation of s = 6.80. Is this result sufficient to conclude that students who do well in school spend significantly less time on online social media? Use a one-tailed test with α = .05.
   

1. 
   State the null hypothesis in words and in a statistical form (1).
   

2. 
   State the alternative hypothesis in words and a statistical form (1).
   

3. 
   Compute the appropriate statistic to test the hypotheses. Sketch the distribution with the estimated standard error and locate the critical region(s) with the critical value(s). (3)
   

4. 
   State your statistical decision. (.5)
   

5. 
   Compute Cohen’s d. Interpret what this d really means in this context. (2)
   

3. Explain why it would not be reasonable to use estimation (95%CI) after a hypothesis test for which the decision was “fail to reject the H₀.” (1)

4. The value of t-score in a hypothesis testing is influenced by a variety of factors. Assuming that all other variables are held constant, explain how the value of t is influenced by each of the following: (1.5)

1. 
   Increasing the difference between sample mean and the original population mean.
   

2. 
   Increasing the sample variance.
   

c. Increasing the number of scores in the sample.

t-test for two Independent Samples (due 3/23/15 – 3/30/15) (30.5 pts)

1. 
   Describe what is measured by the estimated standard error used for the independent-measures t-test? (1)
   

2. 
   If other factors are held constant, explain how each of the following influences the value of the independent-measures t test and the likelihood of rejecting the null hypothesis? (1)
   

1. 
   Increasing the number of scores in each sample
   

3. 
   Describe the homogeneity of variance assumption and explain why it is important for the independent-measures t test? (1)
   

4. 
   The following problems should be computed both by hand and by SPSS (you need to save the data. You will use the same data for the bonus question that appears later).
   

Research has shown that people are more likely to show dishonest and self-interested behaviors in darkness than in a well-lit environment (Zhong, Bohns, & Gino, 2010). In one experiment, participants were give a set of 20 puzzles and were paid$0.50 for each one solved in a 5-minute period. However, the participants reported their own performance and there was no obvious method for checking their honesty. Thus, the task provided a clear opportunity to cheat and receive undeserved money. One group of participants was tested in a room with dimmed lighting and a second group was tested in a well-lit room. The reported number of solved puzzles was recorded for each individual. The following data represent results similar to those obtained in the study.

a. What are IV and DV in this study? (1)

2. 
   Compute the mean and the standard deviation for each condition (show your work). (6)
   

c. Is there a significant difference in reported performance between the two conditions? Use a two-tailed test and α = .05.

2) State the alternative hypothesis in words and a statistical form. (1)

3) Compute the appropriate statistic to test the hypotheses. Sketch the distribution with the estimated standard error and locate the critical region(s) with the critical value(s). (4)

4) State your statistical decision (.5).

6) Is the homogeneity of assumption met? Conduct the F-max test (3).

7) Compute 95% CI (2) .

9) What is your conclusion? Interpret the results and describe in words. Do not forget to include a statistical form (e.g., t-score, df, type of test, α, Cohen’s d) (2).