Intro to Statistics

On average, what value is expected for the t statistic when the null hypothesis is true?

0

If a hypothesis test is found to have power = 0.70, what is the probability that the test will result in a Type II error?

0.30

For a repeated-measures study comparing two treatments with a sample of n = 9 participants, the difference scores have a mean of MD = 4.90 with SS = 72. What is the estimated standard error for the sample mean difference?

1.0

The results of a hypothesis test with the t-test for a single sample are reported as follows: t(15) = 2.70, p < .05. Based on this report, how many individuals were in the sample?

16

A sample of n = 25 scores has a mean of M = 40 and a standard deviation of s = 10. What is the estimated standard error for the sample mean?

2

An independent-measures study uses n = 15 participants in each group to compare two treatment conditions. What is the df value for the t statistic for this study?

28

The following data were obtained from a repeated-measures study of memory performance before and after drinking 2 cups of coffee. What is the value of mean difference MD for these data? Participant Before After #1 10 15 #2 4 8 #3 7 5 #4 6 11

3.0 (or -3.0)

Two samples, each with n = 5 scores, have a pooled variance, sp2 = 40. What is the estimated standard error, s(M1-M2) for the sample mean difference?

4

Two samples, each with n = 9 scores, produce an independent samples t statistic of t(16) = 2.00, p < .05 with one-tail test. If the effect size is measured using r2, what is the value of r2?

4/20

A researcher conducts an independent-measures study examining the effectiveness of a group exercise program at an assisted living facility for elderly adults. One group of residents is selected to participate in the program, and a second group serves as a control. After 6 weeks, the researcher assessed physical fitness of each participant. The data are as follows: Exercise Group: Control Group: n = 15 n = 10 M = 15.5 M = 12 SS = 190 SS = 120.5 A. Does the exercise program have a significant effect on physical fitness? Use p < .05, 2-tails test. B. Compute Cohen's d to measure the size of the treatment effect. Is this a large effect? C. Write a conclusion demonstrating how the outcome of the hypothesis test and the measure of effect size would appear in a research report in APA style.

A. The pooled variance: sp2 = (190 + 120.5/(15-1) + (10-1) = 310.5/23 = 13.5 The estimated standard error: sM1-M2 = √13.5/10 +13.5/15 = √2.25 = 1.5 t = (15.5 - 12)/1.5 = 3.5/1.5 = 2.33 t critical for df = 23 and p < .05 is 2.069 B. Cohen's d = 3.5/√13.5 = 3.5/3.67 = 0.95; this is a large effect. C. The group exercise program had a significant effect on physical fitness of elderly adults, t(23) = 2.33, p < .05, d = 0.95.

A teacher gives a reading skills test to a third-grade class of n = 25 children at the beginning of the school year. To evaluate the changes that occur during the year, students are tested again at the end of the year. Their test scores showed an average improvement of MD = 5.7 points with s2 = 100. A. Are the results sufficient to conclude that there is significant improvement in children's reading skills? Use a one-tailed test with α = .01. B. Compute Cohen's d to measure the size of the effect. Is this a large effect? C. Write a conclusion demonstrating how the outcome of the hypothesis test and the measure of effect size would appear in a research report in APA style.

A. The standard error: Sq.Root of 100/25 = √4 = 2 t(24) = 5.7/2 = 2.85. t critical for df = 24 and p <.01 (1-tail test) is 2.492 B. s = √ s2 = √100 = 10; Cohen's d = MD/s =5.7/10 = 0.57; this is a medium size effect. C. There was a significant improvement in children's reading skills at the end of the school year, t(24) = 2.85, p <.01, d = 0.57.

The term error is used in two different ways in hypothesis testing: a standard error and a Type I error. A. What can a researcher do to influence the size of the standard error? Does this action have any effect on the probability of a Type I error? B. What can a researcher do to influence the probability of a Type I error? Does this action have any effect on the size of the standard error?

A. A standard error can be decreased by increasing the sample size. Changes in sample size have no effect on the probability of a Type I error. B. The probability of a Type I error is determined by selecting the alpha level for the test. The choice of an alpha level does not affect the value of standard error.

A sample is selected from a population with µ = 50. After a treatment is administered to the individuals in the sample, the mean is found to be M = 55 and the variance is s2 = 64. A. If the sample has n = 4 scores, then conduct a hypothesis test to evaluate the significance of the treatment effect and calculate Cohen's d to measure the size of the treatment effect. Use a two-tailed test with α = .05. B. If the sample has n = 16 scores, then conduct a hypothesis test to evaluate the significance of the treatment effect and calculate Cohen's d to measure the size of the treatment effect. Use a two-tailed test with α = .05. C. Describe how increasing the size of the sample affects the likelihood of rejecting the null hypothesis and the measure of effect size.

A. The estimated standard error: SM = SqRoot of s2/ n = SqRoot of 64/4 = 4 t = (55 - 50)/4 = 1.25 Cohen's d = 5/8 = 0.625 With df = 3, the critical t value is t = 3.182. Therefore, fail to reject the null hypothesis. There is no significant effect of treatment, t(3)= 1.25, p > .05. B. The estimated standard error: SM = SqRoot of s2/ n = SqRoot of 64/16 = 2 t = (55 - 50)/2 = 2.50. Cohen's d = 5/8 = 0.625 With df = 15, the critical t value is t = 2.131. Reject the null hypothesis. There is a significant effect of treatment, t(15) = 2.5 ,p < .05, d = 0.625. C. Increasing sample size increases the likelihood of rejecting the null hypothesis but has no effect on the value of effect size (i.e., if the sample is very small even relatively large effect may not be statistically significant).

A sample of n = 16 individuals is selected from a population with µ = 30. After a treatment is administered to the individuals, the sample mean is found to be M = 33. A. If the sample variance is s2 = 16, then conduct a hypothesis test to evaluate the significance of the treatment effect and calculate r2 to measure the size of the treatment effect. Use a two-tailed test with α = .05. B. If the sample variance is s2 =64, then conduct a hypothesis test to evaluate the significance of the treatment effect and calculate r2 to measure the size of the treatment effect. Use a two-tailed test with α = .05. C. Describe how increasing variance affects the likelihood of rejecting the null hypothesis and the measure of effect size.

A. The standard error: SM = SqRoot of s2/ n = SqRoot of 16/16 = 1 t = (33-30)/1 = 3.00. r2 = 9/24 = 0.375 With df = 15, the critical value is t = 2.131. Reject the null hypothesis.There is a significant treatment effect, t(15) = 3.00, p < .05, r2 = 0.375. B. The standard error: SM = SqRoot of s2/ n = SqRoot of 64/16 = 2 t = (33-30)/2= 1.50. r2 = 2.25/17.25 = 0.13 With df = 15, the critical value is t = 2.131. Fail to reject the null hypothesis. There is no evidence of significant treatment effect, t(15) = 1.5, p > .05. C. Increasing sample variance decreases the likelihood of rejecting the null hypothesis and decreases measures of effect size.

The t-test for independent sample can be used to examine ____________.

A.the mean difference between two treatment conditions in an experiment (e.g. a difference in performance of experimental group and control group). B.the mean difference between two populations in quasi-experimental designs (e.g., mean difference in attitudes to abortion between residents of the southern vs. northern states in the U.S.).

What is indicated by a large variance for a sample of difference scores (i.e., a large variance of D scores)?

An inconsistent treatment effect and a low likelihood of a significant difference.

Which of the following accurately describes a hypothesis test statistical method?

An inferential statistical method that uses the data from a sample to draw inferences about a population

What is the relationship between the alpha level, the size of the critical region, and the risk of a Type I error?

As the alpha level increases, the size of the critical region increases, and the risk of a Type I error increases.

For which of the following situations would a repeated-measures research design be appropriate?

Comparing patients' pain tolerance at the beginning and at the end of physical therapy sessions.

Which of the following research situations would be most likely to use a between-subjects research design?

Examining gender differences in analytical problem solving skills among undergraduate students.

A researcher obtains t = 2.25 for a repeated-measures study using a sample of n = 10 participants. Based on this t value, what is the correct decision for a two-tailed test?

Fail to reject the null hypothesis with either α = .05 or α = .01.

Which of the following is an accurate definition of a Type II error?

Failing to reject a false null hypothesis

When n is small (less than 30), how does the shape of the t distribution compare to the normal distribution?

It is flatter and more spread out than the normal distribution.

If other factors are held constant, what is the effect of increasing the sample size? (Hint: Look at the formulas for the estimated standard error and t-test for a single sample to figure out the answer).

It will decrease the estimated standard error and increase the likelihood of rejecting H0.

If other factors are held constant, what is the effect of increasing the sample variance? (Hint: Look at the formulas for the estimated standard error and t-test for a single sample to figure out the answer).

It will increase the estimated standard error and decrease the likelihood of rejecting H0.

In a repeated-measures experiment, each individual participates in one treatment condition and then moves on to a second treatment condition. One of the major concerns in this type of study is that participation in the first treatment may influence the participant's score in the second treatment. What is this problem is called?

Order effect.

Two samples, each with n = 8, produced an independent sample t statistic of t = -2.17. Which of the following the correct decisions for a two-tailed test?

Reject H0 with α = .05 but fail to reject with α = .01.

An independent-measures research study examines gender differences in toddlers' verbal skills. The researchers tested verbal skills of ten 2-years old girls (i.e., n1 = 10) and ten 2-years old boys (i.e., n2 = 10). The data produced a t(18) = 2.35. Which of the following is the correct decision for a two-tailed hypothesis test with p < .05?

Reject the H0 with α = .05; there is a significant gender difference in toddlers' verbal skills.

A two-tailed hypothesis test is being used to evaluate a treatment effect with α = .05. If the sample data produce a z-score of z = - 2.24, what is the correct decision?

Reject the null hypothesis and conclude that the treatment has an effect.

If a hypothesis test produces a z-score in the critical region, what decision should be made?

Reject the null hypothesis.

Which of the following is an accurate definition of a Type I error?

Rejecting a true null hypothesis

Some researchers claim that herbal supplements such as ginseng enhance human memory. To test this claim, a researcher selects a sample of n = 25 college students. Each student is given a ginseng supplement daily for six weeks and then all the participants are given a standardized memory test. For the population, scores on the standardized memory test are normally distributed with μ = 70 and σ = 15. The sample of n = 25 students after receiving the ginseng treatment had a mean memory score of M = 76. A. Do the data indicate a significant effect of the ginseng supplement on memory? Formulate the null and alternative hypotheses, compute the z-test and interpret the z-test outcome to answer this research question. Use a two-tailed test with α = .05 B. Do the supplement show a large effect? Compute Cohen's d to establish the size of the effect.

Some researchers claim that herbal supplements such as ginseng enhance human memory. To test this claim, a researcher selects a sample of n = 25 college students. Each student is given a ginseng supplement daily for six weeks and then all the participants are given a standardized memory test. For the population, scores on the standardized memory test are normally distributed with μ = 70 and σ = 15. The sample of n = 25 students after receiving the ginseng treatment had a mean memory score of M = 76. A. Do the data indicate a significant effect of the ginseng supplement on memory? Formulate the null and alternative hypotheses, compute the z-test and interpret the z-test outcome to answer this research question. Use a two-tailed test with α = .05 B. Do the supplement show a large effect? Compute Cohen's d to establish the size of the effect.

What is measured by the numerator of the z-score test statistic?

The actual distance between a sample mean M and a population mean µ.

What is measured by the denominator of the z-score test statistic?

The average distance between a sample mean M and the population mean µ that would be expected if H0 was true.

Which of the following is an accurate definition for the power of a statistical test?

The probability of rejecting a false null hypothesis

A hypothesis test with a sample of n = 30 participants produces a t statistic of t = 2.33. Assuming a one-tailed test , what is the correct decision about the outcome of the test?

The researcher can reject the null hypothesis with α = .05 but not with α = .01.

A sample of n = 25 scores produces a t statistic of t = 2.05. If the researcher is using a two-tailed test, which of the following is the correct statistical decision?

The researcher must fail to reject the null hypothesis with either α = .05 or α = .01.

A sample of n = 25 individuals is selected from a population with µ = 80, and a treatment is administered to the sample. What is expected if the treatment has no effect?

The sample mean should be close 80 and should lead you to fail to reject the null hypothesis.

Which of the following is a fundamental difference between the t statistic and a z-score?

The t statistic uses the sample variance in place of the population variance.

What value is estimated with a confidence interval using the t statistic?

The value for an unknown population mean

Which of the following is the correct null hypothesis for an independent samples t-test (assume 2-tails test)?

There is no difference between populations represented by two samples (i.e., μ1 - μ2 = 0 or μ1 = μ2).

For which of the following situations would a repeated-measures design have the maximum advantage over an independent-measures design?

When very few subjects are available and individual differences are large.

A sample is selected from a population with μ = 46, and a treatment is administered to the sample. After treatment, the sample mean is M = 48 with a sample variance of s2 = 16. Based on this information, what is the effect size evaluated by Cohen's d?

d = 0.50

A sample of n = 9 individuals is selected from a population with μ = 60 and σ = 6, and a treatment is administered to the sample. After treatment, the sample mean is M = 64. What is the size of the treatment effect evaluated by Cohen's d for this sample?

d = 0.67

The estimated standard error, s(M1-M2) in the t-test for independent samples ________________.

is computed based on the variances, s2 of both samples

A researcher reports t(12) = 2.86, p < .05 for a repeated-measures research study. How many individuals participated in the study?

n = 13

A sample of n = 16 scores produces a t statistic of t = 2.00. Based on this information, what is the effect size evaluated by r2?

r2 = 4/19

What is the sample variance and the estimated standard error for a sample of n = 9 scores with SS = 72?

s2 = 9 and sM = 1

Which of the following accurately describes outcomes in the critical region of the distribution of sample means for z-test?

sample means with a very low probability if the null hypothesis is true.ample means with a very low probability if the null hypothesis is true.

With α = .01, the two-tailed critical region for a t test using a sample of n = 16 subjects would have boundaries of ______.

t = ±2.947

Which of the following is the correct style of reporting the results of a hypothesis test with a t-test for a single sample and a measure of effect size using a t statistic, according to APA style?

t(19) = 2.30, p < .05, r2 = 0.42,

What is the average value expected for the independent-measures t statistic if the null hypothesis is true (i.e., the two samples represent the same population)?

t=0

The statement of the null hypothesis refers to which of the following?

the population after treatment.

If a research report from an experiment investigating the effect of stress on memory states "t(40) = 2.31, p < .05." then _______________.

the researcher rejected H0 and found a significant effect of stress on memory.

A researcher is conducting an experiment to evaluate the effectiveness of treatment for individuals in a population that is known to have a mean of μ = 25. The results will be examined using a two-tailed hypothesis test. Which of the following is the correct statement of the null hypothesis?

μ = 25 (i.e., the treatment has no effect on the population mean).

A researcher selects a sample from a population with a mean of μ = 40 and administers a treatment to the individuals in the sample. Which of the following is the correct statement of the null hypothesis for a two-tailed t-test for a single sample?

μ = 40

A sample of n = 4 scores is selected from a population with an unknown mean. The sample has a mean of M = 40 and a variance of s2 = 16. Which of the following is the correct 90% confidence interval for μ?

μ = 40 ± 2.353 x 2

A researcher administers a treatment to a sample of participants selected from a population with µ = 50. If the researcher obtains a sample mean of M = 55, which combination of factors is most likely to result in rejecting the null hypothesis?

σ = 5 and α = .05