# Two Tailed Test & Repeated Measures Design – Statistics

### Two Tailed Test, Repeated Measures Design, Standard Deviation, The Null Hypothesis – STATS

The key terms in these Statistic chapters include Distribution, Population, Repeated Measures Design, Z-Score, Rejecting the Null Hypothesis, Standard Deviation, Mean, Two Tailed Test – Statistic.

For a population with μ=100 and σ=20, what is the X value corresponding to z=-0.75?

85
x = μ+zσ

Under what circumstances is a score that is located 5 points above the mean a central value, relatively close to the mean?

When the population standard deviation is much greater than 5

On the exam with μ=52, you have a score of x=44. Which value for the standard deviation would give you the highest position in the class distribution?

σ = 8

A population has μ=50. What value of ó would make X=55 an extreme value out in the tail of the distribution?

σ=1
z=x-μ/σ

You have a score of X=65 on the exam. Which set of parameters would give you the best grade on the exam?

μ=60 and σ=5

Last week, Sarah had exams in math and in Spanish. On the math exam, the mean was μ=30 with a ó=5, and Sarah had a score of X=45. On the Spanish exam, the mean was μ=60 with a ó=8, and Sarah had a score of X=68. For

Which class should sara expect the better grade

Math

What is the position in the distribution corresponds to a z-score of z=-1.00?

Below the mean by a distance equal to 1 standard deviation

Which of the following z-score values represents the location farthest from the mean?

#### z=-2.00

For a population with μ=70 and σ=8, what is the z-score corresponding to x=82?

+1.50
z=x-μ/σ

For a population with a standard deviation of σ=10, what is the z-score corresponding to a score that is 5 points below the mean?

Z = -0.50

For a population with μ=100 and σ=20, what is the X value corresponding to z=-0.75?

85
x = μ+zσ

Under what circumstances is a score that is located 5 points above the mean a central value, relatively close to the mean?

When the population standard deviation is much greater than 5

On the exam with μ=52, you have a score of x=44. Which value for the standard deviation would give you the highest position in the class distribution?

σ = 8

A population has μ=50. What value of ó would make X=55 an extreme value out in the tail of the distribution?

σ=1
z=x-μ/σ

You have a score of X=65 on the exam. Which set of parameters would give you the best grade on the exam?

μ=60 and σ=5

For a population with μ=40 and σ=8, what is the z-score corresponding to x=34?

##### -0.75

Which of the following is a requirement for a random sample?

Every individual has an equal chance of selection. The probabilities cannot change during a series of selection.
There must be sampling with replacement

A class consists of 10 males and 30 females. If one student is randomly selected from the class, what is the probability of selecting a male?

10/40

What is a fundamental difference between the t statistic and z-score

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

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

s^2 = 9 and Sm =1 sample variance= SS/n-1 and estimated standard error= under square root (sample variance/ n)

What is the expected value of M?

It is the mean of the distribution of sample means

What is the standard error of M?

It is the standard deviation of the distribution of sample means

For a population with μ=80 and σ=20, the distribution of sample means based on n=16 will have an expected value of ___and a standard error of _______

Expected value of 80 and a standard error of 5 σ/squrt(n)

A sample of n=4 scores is selected from a population with μ=40 and σ=8, and the sample mean is M=43. What is the standard error for the sample mean?

4   Z=m-μ/σm

A sample of n=4 scores is selected from a population with μ=40 and σ=8, and the sample mean is M=43. What is the expected value for the sample mean?

##### 40

What symbol used to identify the standard error of M?

σM

Under what circumstances will the distribution of sample means be normal?

If the population is normal or if the sample size is greater than 30

A random sample of n=60 scores is selected from a population. Which of the following distributions definitely will be normal?

The distribution of sample means will form a normal distribution

A sample of n=36 scores is selected from a population with σ=12. If the sample mean of M=56 produces a z-score of z=+3.00, then what is the population mean

50    Z=sample mean-population mean/standard deviation/squrt(n)

For a normal population with a mean of μ=80 and a standard deviation of σ=10, what is the probability of obtaining a sample mean greater than M=75 for a sample of n=25 scores?

#### P=0.9938

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

Why are t statistics more variable than z-scores?

The extra variability caused by variations in the sample variance

If two samples selected from the same populations, under what circumstances will the two samples have exactly the same t statistic

If the samples are the same size and have the same mean and the same variance

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

0

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

s^2=9 and Sm=1
Sm=Squrt(s^2/n)
S=Squrt(ss/n-1)

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?

#### 2Sm=squrt(s^2/n)

Which set of characteristics will produce the smallest value for the estimated standard error?

A large sample size and a small sample variance

A researcher conducts a hypothesis test using a sample of n=20 from an unknown population. What is the df value for the t statistic?

19

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

A sample of n=25 scores produces a t statistic of t=-2.062. 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 = 0.05 or α = 0.01.

If other factors are held constant, what is the effect of increasing the sample size?

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

Under what circumstances can a very small treatment effect be statistically significant?

If the sample size big and the sample variance is small

How does sample variance influence the estimated standard error and measures of effect size such as r^2 and Cohen’s d?

Larger variance increases the standard error but decreases measures of effect size

Which of the following accurately describes a hypothesis test?

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

A hypothesis test involves a comparison which two elements?

#### Research results from a sample and a hypothesis about a population

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.

For a hypothesis test evaluating the effect of a treatment on a population mean, what basic assumption is made concerning the treatment effect?

If there is a treatment effect, it will add (or subtract) a constant to each score

A researcher selects a sample and administers a treatment to the individuals is the sample. If the sample used for a hypothesis test, what does the null hypothesis (H0) say about the treatment?

The treatment has no effect on the scores.

A sample of n=4 scores has SS=48. What is the estimated standard error for the sample mean?

2 (find the sample variance SS/n-1
estimated standard error= under square root (sample variance/ n) = 2)

A sample of n=4 scores has SS=60 what is the variance for this sample?

20 sample variance = SS/n-1

Why are t statistics more variable than z scores?

The extra variability caused by variations in the sample variance

What is the consequence of a type II error?

Concluding that a treatment has an effect when it really has no effect

When is there a risk of a Type I error?

##### Whenever H0 is rejected

The critical boundaries for a hypothesis test are z=+1.96 and -1.96. If the z-score for the sample data is z=-1.90, what is the correct statistical decision?

Fail to reject H0

A researcher administers a treatment to a sample of participants selected from a population with μ=80. If a hypothesis test is used to evaluate the effect of the treatment, which combination of factors is most likely to result in rejecting the null hypothesis?

A sample mean much different than 80 for a large sample

A researcher expects a treatment to increase the scores for individuals in a population. The treatment is evaluated using a one-tailed hypothesis test, and the test produces z = +2.40. Based on this result, what is the correct statistical decision?

The researcher should reject the null hypothesis with either a = .05 or a = .01

A researcher evaluates a treatment effect using a two-tailed hypothesis test with α= .05, and the decision is to reject the null hypothesis. If the researcher switched to a one-tailed test using the same sample, what decision would be made?

Definitely reject the null hypothesis with alpha = .05 and maybe reject with alpha = .01

If an independent-measures design is being used to compare two treatment conditions, then how many different groups of participants would be needed and how many scores would there be for each participant?

2 groups, 1 score each

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

What is the pooled variance for the following two samples?
Sample 1: n=8 and SS=168
Sample 2: n=6 and SS=120

24

A researcher plans to conduct a research study comparing two treatment conditions with a total of 20 participants.

Which of the following designs would produce 20 scores in each treatment?

A repeated-measures design

A repeated-measures study and an independent-measures study both produced a t statistic with df=10. How many individuals participated in each study?

11 for repeated-measures and 12 for independent measures

A major company would like to assess the impact of using a professional trainer to conduct a confidence-building workshop with its salespeople. A sample of 16 workers is obtained. Half (n=8) attend the workshop and the other hand (n=8) serves as a control group. Two weeks later, each of the participants is given a questionnaire measuring the level of self-confidence.
Controls: M=17.5, SS=100
Workshop: M=22.7, SS=124

Conduct a t-test to determine if there is a difference between the workshop and control conditions: the pooled variance is 16 and the estimated standard error is 2. The t-statistic is 5.2/2=2.6 which is statistically significant (c.v.=2.145)
Find r: r=.57
Use the sample data to construct an 80% confidence interval for the mean difference in confidence between the control group and the workshop conditions: With df=14 and 80% confidence, use t=+1.345. The confidence interval is u1-u2=5.2+1.345(2) and extends from 2.51 to 7.89

For a population with μ=100 and σ=20, what is the X value corresponding to z=-0.75?

#### 85x = μ+zσ

Under what circumstances is a score that is located 5 points above the mean a central value, relatively close to the mean?

When the population standard deviation is much greater than 5

On the exam with μ=52, you have a score of x=44. Which value for the standard deviation would give you the highest position in the class distribution?

σ = 8

A population has μ=50. What value of ó would make X=55 an extreme value out in the tail of the distribution?

σ=1
z=x-μ/σ

You have a score of X=65 on the exam. Which set of parameters would give you the best grade on the exam?

#### μ=60 and σ=5

A researcher would like to evaluate the effectiveness of a pain-relief patch designed for lower back pain. Prior to testing the patch, each of n=8 patients rates the current level of back pain on a scale from 1 to 10. After wearing the patch for 90 minutes, a second pain rating is recorded.
Before: 6,8,9,8,10,5,9,7
After: 2,3,4,1,2,3,8,7

Compute the mean and variance for the sample of difference scores: Md=4 and s^2=8
Do the result indicate a significant difference? Use a two-tailed test: The standard error is 1 and t(7)=4/1=4.00. Reject H0
With df=14 and 80% confidence, use t=+1.345. The confidence interval is u1-u2=-5.2+1.345(2) and extends from 2.51 to 7.89

Which of the following research questions would be appropriate situation for using statistics for using estimation instead of a hypothesis test?

How much improvement will result from a special program to teach reading skills?

Although it is known that a particular therapy is effective for reducing depression, a researcher would like to determine how much effect that therapy has. A sample of depressed patients is obtained and each individual is given a depression test. After four weeks of therapy, their depression is measured again. The scores from the depression test are used to estimate the mean difference in depression. Which estimation equation should the researcher use?

##### The repeated-measure t equation

A researcher knows that a 4 year old girls tend to have better verbal skills than 4 year old boys. To determine how much better, the researcher obtains a sample of boys and a sample of girls and gives each child a verbal ability test.

The scores from the test are used to estimate the mean difference. Which estimation equation should the researcher use?

The independent measures t equation

Which of the following most closely resembles the general form of an interval estimate?

Parameter = statistic + error

The purpose of a confidence interval is to

Use a sample mean or mean difference to estimate the corresponding population mean or mean difference

A sample has a mean of M=53. If this sample is used to make a point estimate of the population mean, then what would the point estimate be?

µ=53

What are the correct values of t to use for a 90% confidence interval for the population mean based on a sample of n=9 scores

#### t=1.860

What value is estimated using the single sample t-statistics

The value for an unknown population mean

What values can be estimate using the independent-measures t statistic

The difference between two population means

What value is estimated using the repeated measures t statistic

The mean for a population of different scores

What is the consequence of a type II error?

Concluding that a treatment has an effect when it really has no effect

When is there a risk of a Type I error?

Whenever H0 is rejected

The critical boundaries for a hypothesis test are z=+1.96 and -1.96. If the z-score for the sample data is z=-1.90, what is the correct statistical decision?

#### Fail to reject H0

A researcher administers a treatment to a sample of participants selected from a population with μ=80. If a hypothesis test is used to evaluate the effect of the treatment, which combination of factors is most likely to result in rejecting the null hypothesis?

A sample mean much different than 80 for a large sample

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