Basic Statistics Final Exam Review

A statistical test used to test the significance of mean difference for two different samples is known as

t-test for independent samples

A statistical test used to test the significance of mean difference (i.e., paired observations) for a single sample is known as

t-test for related samples

For a sample of 50 measurements for an independent-measures design, N1 = 20, N2 = 30, M1 = 15, M2 = 25, pooled variance, sp2 = 182.29. Compute the t-statistic.

t= -2.57

What value of t is required to attain significance for the mean difference of an independent-measures design at the .10 level for a one-tailed test for df = 24?

t= 1.318

What value of t is required to attain significance for the mean difference of paired measurements at the .10 level for a one-tailed test for df = 10?

t= 1.372

For a sample of 26 pairs of measurements, ∑ D=52 and ∑ D^2=400. Compute the t-statistic.

t=2.96

Shape of Frequency Distribution

1. symmetrical 2. positively skewed and negatively skewed

A distribution of scores has a mean of 100 and a standard deviation of 10. For the score 92, what is the corresponding z-score?

-0.8

In a multiple-choice exam, there were 50 questions with 4 possible answers. If a student was guessing at random, what is the probability of getting more than 18 correct just by guessing?

0.0250

For a sample of 20 observations MD = 4, SS = 80. What is the estimated standard error for the sample mean difference?

.459

If you toss a die once, what is the probability of getting a 2, 3, or 4?

.50

A sample n = 25 is selected from a population with mean, μ = 50, σ = 10, what is the probability of obtaining a sample mean that is greater than 55?

0.0062

A sample n = 25 is selected from a population with mean, μ = 70, σ = 10. What is the probability of getting a sample mean that is between 65 and 75?

0.0124

A distribution has mean = 70 and SD = 20. What is the probability of getting a score value between 60 and 75?

0.2902

In an experiment, responses were coded as "yes" or "no." If there are 15 trials or observations, what is the probability of obtaining at leas 9 "yes" responses?

0.3015

A distribution has mean = 50 and SD = 10. What is the probability of obtaining a score value greater than 55?

0.3085

A distribution has mean = 70 and SD = 20. What is the probability of getting a score value that is less than 75?

0.5987

A distribution has mean = 70 and SD = 20. What is the probability of getting a score value that is greater than 60?

0.6915

What value should be obtained when the mean square between treatments is the same as the mean square within treatments?

1.0000

For a sample of 12 observations in an independent-measures study, N1 = 5, N2 = 7, M1 = 6, M2 = 10, sp2 = 8.2. Calculate the standard error of the mean difference.

1.68

Two Types of Statistics

1. Descriptive 2. Inferential

In a certain distribution, there are 1000 women. the probability of pulling the first woman is 1/1000. The probability of obtaining a second woman with sampling with replacement is:

1/1000

Research Designs

1. Experimental 2. Correlation 3. Quasi-Experiment

Characteristics of the Mean

1. If a new score is added to a distribution the mean will change , except when the score is equal to the mean 2. If the value of a score is changed the mean will also change 3. If a constant value is added to the mean, the constant value will be added to the mean 4. If each value in a distribution is multiplied by a constant value, the mean is multiplied by the constant value as well

A researcher concluded the following from a related-samples t-test: The type of illumination has an effect on productivity, t(9) = 6.78, p < .05. How many individuals participated in this study?

10

A population of scores has a mean of 500. If a score value (X) is 750, what value of the population SD would make a score of 750 an extreme score value?

100

What score value (X) corresponds to the top 20% of a distribution with a mean of 80 and SD = 25?

101

In an original distribution, the mean is 50 and the standard deviation is 10. The distribution is then standardized with a new mean of 100 and a standard deviation of 15. For a score of 75 on the original exam, what would the corresponding revised score be in the new standardized distribution?

137.5

For a sample of 12 observations in an independent-measures study, N1 = 5, N2 = 7, SS1 = 64, SS2 = 80. What is the pooled variance, sp2?

14.4

In a distribution of scores, a score value (X) of 515 has a z-score value of 1. The mean for this distribution is 500. What is the standard deviation (SD) of this distribution?

15

A sample of n=37 has a mean of 14; another sample of n=32 has a mean of 18. What is the combined mean for the two samples?

15.86

The mean for a sample of 19 scores is 8. What is ∑X ?

152.0

A population distribution of scores has a standard deviation of 15. What is the z-score for a score 30 points above the mean?

2

For a distribution of scores with a mean of 50, which of the following standard deviations makes 55 a better score (X)?

2

Which of the following values is considered an extreme z-score?

2.15

What z-score value corresponds to a sample mean, M = 43, with a sample size, n = 16, from a population distribution with mean, μ= 40 and σ = 5?

2.4

For a sample of 20 observations MD = 4, s2 = 48. Calculate the t-statistic for related samples.

2.58

For the distribution of scores given in the table below, what is the mean? X : f 5 2 4 4 3 4 2 1 1 5

2.81

A sample is selected from a population with mean, μ= 100 and σ = 15. If the sample mean, M = 106 corresponds to a z-score of 2, how many scores are in the sample?

25

For a sample of 10 observations in each group in an independent-measures study, M1 - M1 = 5, sp2 = 12.5. Calculate the t-statistic for independent samples.

3.16

What is the F-critical value associated with the variation in the data for a comparison of three means for independent samples, each containing 15 cases? Assume alpha level of .05.

3.22

A distribution of sample scores, n = 100, is taken from a population with μ = 500 and σ = 15. If the sample mean is M = 505, what is the corresponding z-score value?

3.33

A sample with n = 36 is obtained from a population with μ = 70, σ = 20. What is the z-score corresponding to the sample mean, M = 73?

3.33

In a t-test for independent samples, a researcher obtained a statistic of 3.35. The researcher used 5 participants in each sample. Assuming an alpha level of .01, two-tailed test, what is the t critical value required for significance?

3.355

For the following set of scores, what is the median?: 4, 6, 7, 2, 1, 3

3.5

In a t-test with related samples a researcher obtained a statistic of 2.35. The researcher used 5 participants in the study. Assuming an alpha level of .01, two-tailed test, what is the t critical value required for significance?

4.604

What is the probability of getting a king in a standard deck of cards?

4/52

How many degrees of freedom are associated with two different samples, with 25 participants in each sample, for a study investigating the effects of illumination of productivity under two treatment conditions?

48

A researcher concluded the following from an independent-samples t-test: The type of illumination has an effect on productivity, t(48) = 7.78, p < .05. How many individuals participated in this study?

50

Given the following: M = 55, standard error = 2, t critical value = 2.064. What is the estimated population mean, considering 95% confidence interval?

50.872, 59.128

In a sample of scores (n=16), the mean is 3. If a constant value of 3 is added to each score, what is the new mean?

6.0

A new distribution was standardized with a mean of 50 and SD = 10. A score in this new distribution is 61. In the original distribution, the mean was 5 with SD = 2. What was the original score value before the distribution was standardized?

7.2

A distribution of scores has a mean of 50 and a standard deviation of 10. For the z-score 3, what is the corresponding raw score value (X)?

80

A sample with n = 70 is selected from a population with μ = 80 and σ = 20. What is the expected value of the mean?

80

What score value (X) corresponds to the top 75% of a distribution with a mean of 100 and SD = 15?

89.95

In a population distribution with mean, μ = 30 and standard deviation, σ = 10, a sample is selected. Which of the following samples is extreme and unlikely representative of the population?

M = 35 and n = 25

A Type I error is:

Rejecting a true null hypothesis

For the table given below, what is the class width? X f 50-54 2 45-49 3 40-44 5 35-39 2 30-34 1

A. 1 B. 6 C. 4 D. 5

For the table given below, what is the total number of scores? Score: f 5 2 4 3 3 1 2 4 1 2

A. 12 B. 35 C. 5 D. 15

For the distribution of quiz scores given below, how many individuals have a score of 4 or below? X : f 5 2 4 3 3 1 2 4 1 2

A. 12 B. it cannot be determined C. 3 D. 10

For the table given below, what is ∑x2 ? Score: f 5 2 4 3 3 1 2 4 1 2

A. 124 B. 12 C. 35 D. 1225

For the following set of scores, what is Σ(X+2)2? Score 2 3 4 5

A. 14 B. 126 C. 54 D. 22

For the table given below, what is ∑X ? Score: f 5 2 4 3 3 1 2 4 1 2

A. 15 B. 5 C. 35 D. 12

For the table given below, how many scores are below 35? X : f 50-54 2 45-49 3 40-44 5 35-39 2 30-34 1

A. 3 B. it cannot be determined C. 1 D. 2

For the following set of scores, what is the 50th percentile? (Hint: construct a frequency distribution and calculate the cumulative percentages.) Score 1 1 2 2 2 2 3 4 4 4 5 5

A. 3.5 B. 2 C. 2.5 D. 3

For the following set of scores, what is Σ(X+2)? Score 2 3 4 5

A. 54 B. 14 C. 2916 D. 22

For the following set of scores, what is ΣX2? Score 2 3 4 5

A. 54 B. 22 C. 14 D. 2916

For the table given below, what percentage of individuals have a score of 4 or below? X: f 5 2 4 3 3 1 2 4 1 2

A. 80% B. 58% C. it cannot be determined D. 83%

Frequency Stem Leaf 2.00 5 02 2.00 6 06 4.00 7 0024 2.00 8 05 Stem width 10 00 Each leaf 1 cases For the above stem-and-leaf display, the most occurring scores are in the:

A. eighties B. fifties C. seventies D. it cannot be determined

For the distribution of quiz scores given below, what proportion of individuals have a score of 4? X: f 5 2 4 3 3 1 2 4 1 2

A. it cannot be determined B. .25 C. .15 D. .75

A researcher is interested in finding out the average age students drink alcohol in the state of Texas. Which of the following holds true?

All students in the state of Texas will be the population under study

For an ANOVA comparing three groups, the null hypothesis states that

All three population means are equal

What is the relationship between standard deviation and standard error?

As the population standard deviation increases, the standard error of the mean increases.

What kind of graph can a researcher use to display data reflecting the assignment of letter grades on a final exam?

Bar Graph

Types of Graphs

Bar graph, polygon, histogram, interval/ratio, stem and leaf display

A researcher obtains data on the sleeping habits for a sample of 50 college students. There were two different groups. In the experiment 25 students from group 1 were asked to sleep for 4 hours a night for one week, and the number of errors made on a task was recorded and group 2 (n = 25) was asked to sleep 8 hours a night for one week, and the number of errors made on a task was recorded. What would be the appropriate design for this study?

Between-subjects design

Which of the following is a limitation using the repeated-measures design?

Carry-over effects

___________ is a statistical procedure that is used to identify a score or value that represents an entire distribution.

Central Tendency

For which of the following situations would an independent-measures study be appropriate?

Comparing social skills for political science majors versus psychology majors

For which of the following situations would a matched-subjects design be appropriate?

Comparing weight loss for participants exercising once a week versus those exercising twice a week

A researcher collects GPA and SAT verbal score data for 1,500 students. She wants to find out if there is a relationship between GPA and SAT verbal score; the research method used to describe and measure this relationship is known as:

Correlational method

Frequency distribution is an example of ___________ Statistics.

Descriptive

Statistical methods used to summarize and describe data are referred to as _______ statistics

Descriptive

A researcher is studying the effects of diet on cholesterol level for a sample of 25 participants. What is the independent variable in this study?

Diet

Confidence intervals are used to

Estimate the population parameters

___________ is used to demonstrate a cause and effect

Experimental Method

ANOVA uses the ______________ to evaluate the mean differences between two or more treatments

F ratio

A researcher obtained the following from One-Way ANOVA: SS between-treatments = 59.34, SS within-treatments = 753.39, and an F-ratio with df = 1, 11. For this analysis, compute the F-ratio. Is the F-ratio significant at alpha=.05

F(1, 11) = .87, p > .05, not significant

For a negatively skewed distribution, the mean would be an appropriate measure of central tendency.

False

The mean, median, and mode are examples of inferential statistics.

False

In a One-Way ANOVA for four groups with 8 cases in each group, the following standard deviations were obtained for each group: S1 = 3, S2 = 3, S3 = 2, and S4 = 5. Perform the F max test for homogeneity of variance at alpha=.05

Fmax = 6.25 < 8.44

Interpretation

For a given data value, the z-score gives the number of standard deviations above (positive) or below (negative) the mean.

IQR

Interquartile Range

In order for you to use a histogram to display your data, the variable must be measured on:

Interval or ratio scale of measurement

A researcher is analyzing her data and wants to compute the mean GPA as an average for the data set. For her data set, the GPA ranges from 1 to 4, with intermediate values in between. What scale of measurement did she use for GPA?

Interval/ratio scale

In a distribution of scores, a score value (X) has a z-score of 2. How would we interpret this score?

It is two standard deviations above the mean.

How do sample size and sample variance influence the likelihood of rejecting the null hypothesis?

Larger sample size and smaller variance increase the likelihood of rejecting the null hypothesis.

For a sample of 5 observations ∑D2=200. Calculate the mean for the difference scores.

MD = 4

M

Mean

The following distribution of sample scores: 2, 2, 3, 5, has been transformed into z-scores. What is the mean and standard deviation for the z-scores?

Mean = 0, SD = 1

In a population, the mean (μ) = 50 with σ =10. If n = 25 scores are selected from the population, what are the expected value of the mean and the standard error of the mean?

Mean = 50; Standard Error = 2

An instructor gave a statistics midterm exam to 50 students, and found that one of the students' scores was note recorded. What measure of central tendency would you use to calculate the average for the 50 students who took the exam?

Median

Median

Midpoint of a distribution

For a sample of 42 observations in an independent-measures study, N1 = 24, N2 = 18, M1 - M2 = 3 and sp2 = 112.50. Is the difference between the means significant at .05 level for a two-tailed test?

NO, t = .907, df = 40, p > .05

Your house number on the street you live on would be an example of

Nominal scale of measurement

Statistic

Numerical value that describes a sample example: average GPA for a sample of 500 students

Parameters

Numerical value that describes population example: average GPA for entire student body

A statistical procedure for testing the significance of the difference between two or more means is known as

One-way ANOVA

Assignment of letter grades on a final exam would be an example of using a(n) __________ scale of measurement

Ordinal

A(n) __________________ is the average or a value obtained from a population

Parameter

The likelihood of obtaining a particular event is known as

Probability

The relationship between a sample and a population is based on

Probability

A measure of reaction time would be an example of

Ratio scale of measure

A researcher performed a hypothesis test to evaluate the difference between a sample mean and a population mean and reported these values: t (24) = 2.75, p < .05. Should you reject the null hypothesis, assuming a two-tailed test with alpha level of .05?

Reject the null hypothesis because the difference is significant.

S^2

Sample Variance

Which of the following will produce the smallest value for the standard error of the mean?

Sample size, n = 25 and population SD, σ = 5

_______ is the amount of difference between a parameter and a statistic

Sampling error

S

Squared Deviation

Which of the following statements describe a Type II error?

Stating that there was no effect when in fact there was an effect.

Central Tendency

Statistical procedure that uses a single score value to represent an entire distribution

John Smith, an investigator wishes to calculate the average GPA for a sample of 50 students, calculates this average to be 2.85. In statistical terms, this average is known as a(n):

Statistics

Last week, Mary took a statistics exam, and her score (X) was 55. For this distribution of scores, the mean was 50 and the standard deviation was 2. The same week, she took a biology exam, and her score (X) was 60. For this distribution of scores, the mean was 50 with a standard deviation of 10. When comparing both exams, Mary scored better in:

Statistics

Which of the following is a limitation using the independent-measures design?

The design uses many subjects

The null hypothesis for independent-samples t-test states that

The difference in means for the two populations is zero

For the following set of scores, which holds true?: 1, 2, 3, 4, 4, 10, 15.

The mean has a higher value than the median, and this is a positively skewed distribution

What is the difference between the null hypothesis and the alternative hypothesis?

The null hypothesis is a statement of no difference, while the alternative hypothesis is a statement of difference.

The null hypothesis for a related-samples t-test states that

The population mean difference is zero

Why do researchers state hypotheses in population parameters?

The researcher is often interested in the population.

For a skewed distribution of scores, the median is the preferred measure of central tendency.

True

Scores by themselves are meaningless without additional information.

True

The median is also known as the 50th percentile.

True

The mode is the preferred measure of central tendency for a nominal or discrete variable.

True

The most commonly-used measure of central tendency is the mean.

True

In comparing more than two independent sample means, multiple t-test will result in:

Type II error

The critical values or critical regions define the extreme sample data that are

Unlikely to occur if the null hypothesis is true.

Under what circumstances would a binomial distribution approximate a normal distribution?

When both pn and qn are greater than or equal to 10

What is the relationship between sample size and standard error of the mean?

When sample size (n) increases, standard error of the mean also increases

A researcher obtains data on the sleeping habits for a sample of 50 college students. In one experiment, students were asked to sleep for 4 hours a night for one week, and the number of errors made on a task was recorded and in the following week, the same group of students were asked to sleep 8 hours a night, and the number of errors made on a task was recorded. What would be the appropriate design for this study?

Within-subjects design (Repeated-measures design)

An investigator calculated an F statistic to be F = 23.15. There were five independent samples being compared, with 5 cases in each sample. Assuming an alpha level of .01, are the five means significantly different?

YES, p < .01, the F critical value is 4.43

For a sample of 20 observations MD = 4 and s2 = 4.21 Is the difference between the means significant at .05 level for a two-tailed test?

YES, t = 8.71, df = 19, p < .05

In a t-test for independent samples, a researcher obtained a statistic of 4.52. The researcher used 15 participants in each sample. Assuming an alpha level of .05, two-tailed test, should the researcher reject the null hypothesis?

YES; p<.05

In a t-test with related samples a researcher obtained a statistic of 5.79. The researcher used 7 participants in the study. Assuming an alpha level of .05, two-tailed test, should the researcher reject the null hypothesis?

YES; p<.05

Which descriptive measure is used to find the exact location of a raw score within a distribution?

Z-score

Z

Z-value given

Graph

can be used to visually inspect data

Nominal Scale

classification of observations into categories

Sample

collect data from a defined sample, n=100

Distribution of Sample Means

collections of means from all the possible random samples of a specific size (n) taken from a specific population

Experimental

demonstrate cause and effect 1. manipulation 2. control 3. random assignment

Correlation Design

describes and measures relationship between two variables

How many degrees of freedom are associated with a sample of n = 25 for investigating the effects of illumination of productivity under two treatment conditions?

df = 24

Complete the following ANOVA table: Source SS df MS Between 15 ____ 5 F = ____ Within 20 10 ____

df between = 3, MS within = 2, F = 2.5

Population

make statistical inference or inductive inference about the population

How many degrees of freedom are associated with the variation in the data for a comparison of three means for independent samples, each containing 15 cases?

dfbetween-treatments = 2, dfwithin-treatments = 42

Interval Scale

equal intervals between two values

Ratio Scales

equal intervals between two values and absolute zero

Which of the following is not an assumption underlying the analysis of variance (ANOVA)?

equality of sample means

An independent variable in ANOVA is referred to a

factor

The knowledge of the scales of measurement is important because it enables the researcher to:

identify the appropriate statistic for data analysis

Positively Skewed

mean value higher than the mean; mode has the lowest score value

Negatively Skewed

mean value is lower than the median; mode has the highest score value

Symmetrical

mean, median, and mode in a symmetrical distribution have equal values

Quasi-Experiment

missing requirements from an experiment

Mean

most commonly used measure of central tendency; arithmetic measure

Z-Scores

numerical value that identifies the precise location of a raw score (x) within a distribution

Descriptive Statistics

organize, summarize, and describe data examples: frequency distribution, measures of central tendency (mean, median, mode), and measures of variability (range, variance, and standard deviation)

A sample n = 25 is selected from a population with mean, μ = 50, σ = 10, a researcher calculated the sample mean, M = 55. Assuming the researcher was performing a two-tailed test, is the sample mean significantly different from the population mean? For the test, α = .05.

p = 0.0124, significant difference between the sample mean and the population mean

In a population of scores, a particular score is 5 points above the mean. This particular score will be a central value when the:

population standard deviation is greater than 5

Ordinal Scale

ranking of observations

X

raw score to be determined

Percentile Ranks

refers to a percentage of individuals or observations falling below a given score value

Percentile

refers to a score value

Mode

score value with the greatest frequency

Statistics

set of mathematical procedures for the analysis and interpretation of data

O

standard deviation given

Variability

the degree to which scores in a data set are spread out or close together (clustered)

Sample and Error

the difference between parameters and statistics

Probability

the frequency of times an outcome occurs divided by the total number of possible outcome(s)

The expected value of the mean is

the mean of the distribution of sample means

The alpha level is the criterion for evaluating

the null hypothesis

In the sample size n = 20, under what condition will the distribution of sample means be a normal distribution?

the population is normally distributed

The standard error of the mean can be defined as

the standard deviation of the distribution of sample means

A z value calculated for a hypothesis test is known as

the test for the statistic

Tables

used to organize data tables and can be constructive with ungrouped and grouped data with columns for a score value (x), frequency, proportion, and cumulative percentages (%)

Inferential Statistics

uses sample data to draw conclusions about a population examples: z-scores, t-tests, ANOVA

The F statistic is a ratio of two

variances

The purpose of post-hoc procedures is to determine

which treatments or groups are significantly different

A sample with n = 36 is obtained from a population with mean, μ = 70, σ = 20. The sample mean, M = 73. Is the sample mean significantly different from the population mean? Test at the .01 level of significance.

z = 0.90, the sample mean is not significantly different from the population mean, p > .01

In a population of scores, a particular raw score is 10 points above the mean with SD = 10. What is the z-score value of this raw score, and what is the probability of getting a z-score beyond this value?

z = 1, p = 0.1587

In a certain population, an investigator predicted that the sample mean, M = 495 was significantly lower than the population mean, μ = 500, with a standard deviation, σ = 15. The sample size was 25. Is the investigator correct to say that the sample mean was significantly lower than the population mean? Make a decision, test at .05 level of significance.

z = 1.67, p = 0 .0475, accept the investigator's statement

Which of the following statements is a definition of hypothesis?

→Potential relationship between two or more variables →Statement that can be tested →Predicted outcome

Scales of Measurement

→nominal scale → ordinal scale → interval scale → ratio scale

Statistical Notation

∑ summation, "sum of", "add" ∑X ∑x2 ∑(x-2)