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Mathematics > Statistics

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Chapter 9: Evens

2. A sample of n =25 scores has a mean of M = 83 and a standard deviation of s = 15.

a. Explain what is measured by the sample standard deviation.

b. Compute the estimated standard error for the sample mean and explain what is measured by the standard error.

4. Explain why t distributions tend to be flatter and more spread out than the normal distribution.

6. The following sample of n = 6 scores was obtained from a population with unknown parameters. Scores: 7, 1, 6, 3, 6, 7

a. Compute the sample mean and standard deviation. (Note that these are descriptive values that summarize the sample data.)

b. Compute the estimated standard error for M. (Note that this is an inferential value that describes how accurately the sample mean represents the unknown population mean.)

8. To evaluate the effect of a treatment, a sample is obtained from a population with a mean of μ = 75, and the treatment is administered to the individuals in the sample. After treatment, the sample mean is found to be M =79.6 with a standard deviation of s = 12.

a. If the sample consists of n = 16 individuals, are the data sufficient to conclude that the treatment has a significant effect using a two-tailed test with α =.05?

b. If the sample consists of n = 36 individuals, are the data sufficient to conclude that the treatment has a significant effect using a two-tailed test with α =.05?

c. Comparing your answer for parts a and b, how does the size of the sample influence the outcome of a hypothesis test?

10. A random sample of n = 16 individuals is selected from a population with μ = 70, and a treatment is administered to each individual in the sample. After treatment, the sample mean is found to be M = 76 with SS = 960.

a. How much difference is there between the mean for the treated sample and the mean for the original population? (Note: In a hypothesis test, this value forms the numerator of the t statistic.)

b. How much difference is expected just by chance between the sample mean and its population mean? That is, find the standard error for M. (Note: In a hypothesis test, this value is the denominator of the t statistic.)

c. Based on the sample data, does the treatment have a significant effect? Use a two-tailed test with α =.05.

12. Many animals, including humans, tend to avoid direct eye contact and even patterns that look like eyes. Some insects, including moths, have evolved eye-spot patterns on their wings to help ward off predators. Scaife (1976) reports a study examining how eye-spot patterns affect the behavior of birds. In the study, the birds were tested in a box with two chambers and were free to move from one chamber to another. In one chamber, two large eye-spots were painted on one wall. The other chamber had plain walls. The researcher recorded the amount of time each bird spent in the plain chamber during a 60-minute session. Suppose the study produced a mean of M = 37 minutes in the plain chamber with SS = 288 for a sample of n = 9 birds. (Note: If the eye-spots have no effect, then the birds should spend an average of

μ = 30 minutes in each chamber.)

a. Is this sample sufficient to conclude that the eyespots have a significant influence on the birds’ behavior? Use a two-tailed test with

α = .05.

b. Compute the estimated Cohen’s d to measure the size of the treatment effect.

c. Construct the 95% confidence interval to estimate the mean amount of time spent on the plain side for the population of birds.

14. The librarian at the local elementary school claims that, on average, the books in the library are more than 20 years old. To test this claim, a student takes a sample of n = 30 books and records the publication date for each. The sample produces an average age of M = 23.8 years with a variance of s2 = 67.5. Use this sample to conduct a one-tailed test with α = .01 to determine whether the average age of the library books is significantly greater than 20 years (μ > 20).

16. In a classic study of infant attachment, Harlow (1959) placed infant monkeys in cages with two artificial surrogate mothers. One “mother” was made from bare wire mesh and contained a baby bottle from which the infants could feed. The other mother was made from soft terry cloth and did not provide any access to food. Harlow observed the infant monkeys and recorded how much time per day was spent with each mother. In a typical day, the infants spent a total of 18 hours clinging to one of the two mothers. If there were no preference between the two, you would expect the time to be divided evenly, with an average of μ = 9 hours for each of the mothers. However, the typical monkey spent around 15 hours per day with the terry-cloth mother, indicating a strong preference for the soft, cuddly mother. Suppose a sample of n = 9 infant monkeys averaged M = 15.3 hours per day with SS = 216 with the terry-cloth mother. Is this result sufficient to conclude that the monkeys spent significantly more time with the softer mother than would be expected if there were no preference? Use a two-tailed test with α = .05.

18. Other research examining the effects of preschool childcare has found that children who spent time in day care, especially high-quality day care, perform better on math and language tests than children who stay home with their mothers (Broberg, Wessels, Lamb, & Hwang, 1997). Typical results, for example, show that a sample of n = 25 children who attended day care before starting school had an average score of M = 87 with SS = 1536 on a standardized math test for which the population mean is μ = 81.

a. Is this sample sufficient to conclude that the children with a history of preschool day care are significantly different from the general population? Use a two-tailed test with α = .01.

b. Compute Cohen’s d to measure the size of the preschool effect.

c. Write a sentence showing how the outcome of the hypothesis test and the measure of effect size would appear in a research report.

20. A random sample is obtained from a population with a mean of μ = 70. A treatment is administered to the individuals in the sample and, after treatment, the sample mean is M = 78 with a standard deviation of s = 20.

a. Assuming that the sample consists of n = 25 scores, compute r2 and the estimated Cohen’s d to measure the size of treatment effect.

b. Assuming that the sample consists of n = 16 scores, compute r2 and the estimated Cohen’s d to measure the size of treatment effect.

c. Comparing your answers from parts a and b, how does the number of scores in the sample influence the measures of effect size?

22. In studies examining the effect of humor on interpersonal attractions, McGee and Shevlin (2009) found that an individual’s sense of humor had a significant effect on how the individual was perceived by others. In one part of the study, female college students were given brief descriptions of a potential romantic partner. The fictitious male was described positively as being single, ambitious, and having good job prospects. For one group of participants, the description also said that he had a great sense of humor. For another group, it said that he had no sense of humor. After reading the description, each participant was asked to rate the attractiveness of the man on a seven-point scale from 1 (very attractive) to 7 (very unattractive). A score of 4 indicates a neutral rating.

a. The females who read the “great sense of humor” description gave the potential partner an average attractiveness score of M = 4.53 with a standard deviation of s = 1.04. If the sample consisted of n = 16 participants, is the average rating significantly higher than neutral (μ = 4)? Use a one-tailed test with α = .05.

b. The females who read the description saying “no sense of humor” gave the potential partner an average attractiveness score of M = 3.30 with a standard deviation of s = 1.18. If the sample consisted of n = 16 participants, is the average rating significantly lower than neutral

(μ _ 4)? Use a one-tailed test with α = .05.

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A sample of n =25 scores has a mean of M = 83 and a standard deviation of s = 15. a. Explain what is measured by the sample standard deviation. b. Compute the estimated standard error for the sample mean and explain what is measured by the standard error.

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Explain why t distributions tend to be flatter and more spread out than the normal distribution.