Exam 2 (Ch. 4, 5, 8)

For a population with μ = 80 and σ = 6, what is the z-score corresponding to X = 68?

-2.00

the deviation scores are calculated for each individual in a population of N = 4. The first three individuals have deviations of 2, 3, -1. What is the deviation for the fourth individual?

-5 The deviation scores for the entire set must add up to zero. The first three deviations add up to 5, so the 4th deviation must be -5.

Increasing Variability leads to:

-larger standard error -smaller z-score (closer to zero) -lower likelihood finding significant treatment effect

Increasing Sample Size leads to:

-smaller standard error -larger z-score -greater likelihood of finding a significant treatment effect -ex: n=25 and n=100, the z-score doubles

For any distribution, what is the z-score corresponding to the mean?

0

What is the standard deviation for the following set of N = 5 scores: 10, 10, 10, 10, and 10.

0 Because there is no variability (the scores are all the same)

If an entire population with μ = 60 and σ = 8 is transformed into z-scores, then the distribution of the z-scores will have a mean of __ and a standard deviation of __.

0; 1

A sample of n = 20 scores is transformed into z-scores. What is the value of SS, the sum of squared deviations, for the set of 20 scores?

19 SSz(squared) = SS/n-1 1 = SS/n-1 n-1 = SS 20 -1 = 19

In a population with a standard deviation of σ = 5, a score of X = 44 corresponds to a z-score of z= 2.00. What is the population mean?

34

For a sample with M = 50 and s = 12, what is the X value corresponding to z = -0.25?

47

For a population with μ = 40 and σ = 8, what is the X value corresponding to z = 1.50?

52

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

85

For a sample with s = 12, a score of X = 73 corresponds to z = -1.00. What is the sample mean?

85

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 -σ(m) = σ/√n To reject null: -want small σ(m) -want large √n

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 with an α = .05

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

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

Below the mean by a distance equal to 1 standard

With α = .05, how are the boundaries for the critical region determined?

Boundaries are drawn so there is 2.5% (0.025) in each tail of the distribution

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 α = .05 and maybe reject with α = .01

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

Fail to reject the null hypothesis

True or False: If a sample mean is in the critical region with α = .05, it would still (always) be in the critical region were changed to α = .01?

False

True or False: If the alpha level is increased from α = .01 to α = .05, then the boundaries for the critical region move farther away from the center of the distribution.

False. The larger alpha means that the boundaries for the critical region move closer to the center of the distribution

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 sample of n = 30 scores, X = 45 corresponds to z= 1.50 and X = 40 corresponds to z= 1.00. What are the values for the sample mean and standard deviation?

M = 30 and s = 10

For an exam with a mean of M = 74 and a standard deviation if s = 8, Mary has a score of X = 80, Bob's score corresponds to z = 1.50, and Sue's score is located above the mean by 10 points. If the students are placed in order from smallest score to largest score, what is the correct order?

Mary, Sue, Bob

Last week Tim got a score of X = 54 on a Math exam with μ = 60 with a σ = 8. He also got a X = 49 on an English exam with μ = 55 with a σ = 3, and he got X = 31 on a psychology exam with μ = 37 with a σ = 4. For which class should Tim expect the best grade?

Math

Last week, Sarah had exams in both Math and Spanish. On the Math exam, the mean was μ = 30 with a σ = 5, and Sarah had a score of X = 45. On the Spanish exam, the mean was μ = 60 with a σ = 8, and Sarah had a score of X = 68. For which class should Sarah expect a better grade?

Math -Sarah was 3 SD higher than the mean in Math. In Spanish she was only 1 SD above

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

Might reject the null hypothesis with α = .05 but not

Which of the following accurately describes the critical region?

Outcomes with a very low probability if the null hypothesis is true (-Reject null if p < α ??)

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

Reject the null hypothesis

A two-tailed hypothesis test is being used to evaluate a treatment effect with α = .05. If a 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

A researcher conducts a hypothesis test to evaluate the effect of treatment that is expected to increase scores. The hypothesis test produces a z-score of z= 2.27. If the researcher is using a one-tailed test, what is the correct statistical decision?

Reject the null hypothesis with α = .05 but not with α = .01

A hypothesis test involves a comparison of which two elements?

Research results from a sample and a hypothesis about a population

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

The average distance between M and μ that would be expected if the null hypothesis was true

Which of the following is directly addressed by the null hypothesis?

The population after treatment ??

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 a z-score of z= 2.40. Based on this result, what is the correct statistical decision?

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

A researcher conducts a hypothesis test to evaluate the effect of a treatment. The hypothesis test produces a z-score of z+ 2.37. Assuming that the researcher is using a two-tailed test, what decision should be made?

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

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

The treatment causes a change in the scores

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

The treatment has no effect on the scores -M is close to μ

True or False: If a sample mean is in the critical region with α = .01, it would still (always) be in the critical region were changed to α = .05?

True

True or False: A small value (near 0) for the z-score statistic is evidence that the sample data are consistent with the null hypothesis.

True A z-score near zero indicates that the data support the null hypothesis

True or False: A z-score value in the critical region means that you reject the null hypothesis

True A z-score value in the critical region means that the sample is not consistent with the null hypothesis

T or F: increasing sample size increases likelihood of rejecting null

True larger small produces smaller standard error, which leads to larger z-score

Under what circumstances is a score that is 15 points above the mean an extreme score relatively far from the mean?

When the population standard deviation is much smaller than 15

Under what circumstances is a Type II error likely to occur?

When the treatment effect is very small

When would you use a one-tailed test?

When you expect the data to go one way or the other The theory has a directional bias

What is the advantage of having a mean of µ = 0 for a distribution of z-scores?

With a mean of zero, all positives scores are above the mean and all negatives scores are below the mean

Explain why the formula for sample variance divides SS by n-1 instead of dividing by n

Without some correction, sample variability consistently underestimates the population variability. Dividing by a smaller number (n-1 instead of n) INCREASES the value of the SAMPLE VARIANCE and makes it an UNBIASED estimate of population variance

Describe the location in the distribution for each of the following z-scores. (For example, z=1.00 is located above the mean by 1 SD) a) z= -1.50 b) z= 0.25 c) z= -2.50 d) z= 0.50

a) above the mean by 1.5 SD b) above the mean by 1/4 SD c) below the mean by 2 &1/2 SD d) above the mean by 1/2 SD

Identify the z-score corresponding to each of the following locations in a distribution a) below the mean by 2 SD b) above the mean by 1/2 SD c) below the mean by 1.5 SD

a) z= -2.00 b) z= 0.50 c) z= -1.50

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

a) z= 0.50 b) z= 1.00 c) z= -1.00 D) z= -2.00 (correct)

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

a) μ = 60 and σ = 10 B) μ = 60 and σ = 5 (correct) c) μ = 70 and σ = 10 d) μ = 70 and σ = 5 -small difference from the mean and small standard deviation

A population has a μ = 70 and σ = 5. a) If 10 points were added to every score, what would be the new values for the mean and standard deviation? b) if every score in the population was multiple by 2, what would be the new values for the mean and standard deviation?

a) μ = 80 and σ = 5 -adding a constant does not change SD b) μ = 140 and σ = 10. -SD is multiple by the same amount

On an 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?

a) σ = 2 b) σ = 4 C) σ = 8 (correct) d) cannot determine from info given -larger standard deviation gives you a better score

significant if...

it is very unlikely to occur when the null hypothesis is true

extreme values

not consistent with null hypothesis reject null hypothesis

Type II error:

occurs when a researcher fails to reject a null hypothesis that is really false the hypothesis test has failed to detect a real treatment effect often happens when the effect of the treatment is relatively small

Type I error:

occurs when a researcher rejects a null hypothesis that is actually true researcher concludes that a treatment does have an effect, when in fact, it has no effect

Critical regions values

provide evidence treatment has an effect reject null hypothesis (that treatment has no effect)

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

the actual distance between M and μ

Calculate the variance for the following population of N = 5 scores: 4, 0, 7, 1, 3

variance = 6 The sum of squared deviations is 30 30/5 = 6

A sample of n = 20 scores has a mean of M = 45 and a standard deviation of s = 8. In this sample, what is the z-score corresponding to X = 57?

z = 1.50

A researcher is conducting an experiment to evaluate a treatment that is expected to increase the scores for individuals in a population. If the researcher uses a one-tailed test with α = .01, then which of the following correctly identifies the critical region?

z > 2.33

larger alpha

z-score = closer to zero

If a researcher conducted a hypothesis test with an alpha level of α = .02, what z-score values would form the boundaries of the critical region?

z-score boundaries would be z= 2.33 and z = -2.33 the .02 would be split between the 2 tails, with .01 in each tail

In a sample with M = 40 and s = 8, what is the z-score corresponding to X = 38?

z= -0.25

For a population with a standard deviation of σ = 6, what is the z-score corresponding to a score that is 12 points above the mean?

z= 2

z score

z= sample mean -(minus) hypothesized population mean/ (divided by) standard error between M and μ

In a population of scores X = 44 corresponds to z= 0.50 and X = 50 corresponds to z= 2.00. What are the values for the population mean and standard deviation?

μ = 42 and σ = 4

A researcher is conducting an experiment to evaluate a treatment that is expected to increase the scores for individuals in a population that is known to have a mean of μ = 80. The results will be examined using a one-tailed hypothesis test. Which of the following in the correct statement of the null hypothesis?

μ ≤ 80

A population has μ = 50. What value of σ would make X = 55 a central, representative score in the population?

σ = 10

are you more likely to reject null hypothesis with a SD σ = 2 or σ = 10?

σ = 2 Smaller SD produces smaller SE which leads to larger z-score

A population has μ = 80. In this population, a score of X = 86 corresponds to z= 2.00. What is the population standard deviation?

σ = 3

A researcher administers a treatment to a sample of participants selected from a population with µ = 80. If the researcher obtains a sample mean of M = 88, which combination of factors is most likely to result in rejecting the null hypothesis?

σ = 5 and n= 50 Reject null: -smaller SD -larger n

A researcher administers a treatment to a sample of participants selected from a population with μ = 80. If the researcher obtains a sample mean of M = 88, which combination of factors is most likely to result in rejecting the null hypothesis?

σ = 5 and α = .05 -want critical region close to μ