Statistics Exam – Frequency & Standard Deviation

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Standard Deviation, Frequency Distribution Table, Hypothesis Testing – Statistics Exam

The key terms in these Statistic chapters include Sample, Compare And Contrast, Frequency Distribution Table, Population, Mode, Random Sampling, Floor Effect, Sample of Scores, Ceiling Effect, Mean, Standard Deviation, Variable, Frequency, Median, Hypothesis Testing, Statistics Exam


The following list of scores is the time (in minutes) that participants were late for statistics exam.
3, 7, 6, 5, 5, 9, 6, 4, 6, 8, 10, 2, 7, 4, 9, 5, 6, 3, 8
Construct a frequency distribution table

Scores – 2, 3, 4, 5, 6, 7, 8, 9, 10
Frequency – 1, 2, 2, 3, 4, 2, 2, 2, 1


For the following sample of scores: 1, 1, 2, 2, 2, 2, 4
a. find the mean, median, and mode
b. if the new individual with the score of X = 2 is added to the sample, what happens to the values for the mean, the median, and the mode

A. mean = 2
median = 2
mode = 2
b. all stay the same


Compare and contrast random sampling and random assignment

Random sampling is when every individual in the population has an equal opportunity/chance of being chosen. Random assignment is once the sample has already been selected, they are randomly placed into the different groups being studied. These both minims bias during hypothesis testing and experimentation


Describe the the of errors that can occur in hypothesis testing and possible reasons for each

Two errors could occur during hypothesis testing. Type 1 occurs when a null hypothesis is rejected but no change occurred. This could be due to errors in the sample that made it seem like change occurred when it didn’t. A larger sample size could help. Type 2 occurs when the null hypothesis is failed to be rejected when change occurred. This could be due to the trail being ran incorrectly or a biased sample.

Frequency distribution

One method for simplifying and organizing data
showing exactly how many individuals are located in each category on the scale of measurement
presents an organized picture of the entire set of scores and it shows where each individual is located relative to others in the distribution


Four different way to visually describe just one variable

Frequency table, grouped frequency table, frequency histograms, frequency polygon

Frequency table

visual depiction of data that shows how often each value occurs – how many scores were at each value
values are listed in one column and the numbers of individuals with scores at that value are listed in the second column


Ungrouped frequency distribution

Is a count of how often each individual value of a variable occurs in a set of data
Is used when the values a variable can take are limited
Biggest value goes on top row
Only calculated for data that have an order, where umbers tel direction (ordinal, interval, ratio-level data)


Grouped frequency distributions

Count how often the values of a variable, grouped into intervals, occur in a set of data
for variables measured at the ordinal level or higher (make intervals that a 5, 10, 20, 25, or 100 units wide, are same width, don’t overlap)


Discreet numbers

Answer the question “how many”, take whole number values, and have no “in-between”

The following list of scores is the time (in minutes) that participants were late for statistics exam.
3, 7, 6, 5, 5, 9, 6, 4, 6, 8, 10, 2, 7, 4, 9, 5, 6, 3, 8 Construct a frequency distribution table

Scores – 2, 3, 4, 5, 6, 7, 8, 9, 10
Frequency – 1, 2, 2, 3, 4, 2, 2, 2, 1


For the following sample of scores: 1, 1, 2, 2, 2, 2, 4
a. find the mean, median, and mode
b. if the new individual with the score of X = 2 is added to the sample, what happens to the values for the mean, the median, and the mode

A. mean = 2
median = 2
mode = 2
b. all stay the same


Compare and contrast random sampling and random assignment

Random sampling is when every individual in the population has an equal opportunity/chance of being chosen. Random assignment is once the sample has already been selected, they are randomly placed into the different groups being studied. These both minims bias during hypothesis testing and experimentation


Describe the the of errors that can occur in hypothesis testing and possible reasons for each

Two errors could occur during hypothesis testing. Type 1 occurs when a null hypothesis is rejected but no change occurred. This could be due to errors in the sample that made it seem like change occurred when it didn’t. A larger sample size could help. Type 2 occurs when the null hypothesis is failed to be rejected when change occurred. This could be due to the trail being ran incorrectly or a biased sample.

Continuous numbers

Answer the question “how much” and can have “in-between” values; specificity of the number, the number of decimal places reported, depends on the precision of the measuring instrument


Bar graphs

For discrete data (nominal or ordinal) that use the heights of vars to indicate frequency, bars do NOT touch


Histograms

For continuous data (interval or ratio level), displayed in graph form, using heights of bars indicate frequency, bars touch


Frequency polygons

For continuous data (interval or ratio), displayed in graphic format, using line connecting dots above interval midpoints, that indicate frequency

Normal distribution

Bell shaped, symmetric, unimodal
Specific frequency distribution


Skewed distribution

In which one of the tails of the distribution is pulled away from center
Extreme values


Positive skewed

Represents floor effects
All data on left, outliers pulling tail to right
Mean>median


Negative skewed

May represent ceiling effects
All data on right, outliers pulling tail to left
Mean<median


Floor effect

A situation in which a constraint prevents a variable from taking values below a certain point

Ceiling effect

A situation in which a constraint prevents a variable from taking on values above a give number

N or n

Number of scores in a data set
N = population
n = sample


E

Summing of set of values


X and/or Y

Individual measurements or scores obtained for a research participant


Order of operations

P – Parenthesis
E – exponents
M – multiply
D – divide
A – add
S – subtract


Central tendency

Determines a single value that accurately describes the center of the distribution and represents the entire distribution of scores


Measurements of central tendency

Mean, median, mode

For a sample with M = 66, a score of X = 55.5 corresponds to z = -1.50. What is the sample standard deviation?

7


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

= 6 (highest score)


A set of scores ranges from a low of X = 25 to a high of X = 33 and has a mean of 29. Which of the following is the most likely value for the standard deviation for these scores

2 points


A sample of n = 6 scores has SS = 40. If these same scores were a population, then the SS for the population would be _____.

40

A population of N = 5 scores has a mean of µ = 20 and a standard deviation of σ = 4. Which of the following is the correct interpretation of the standard deviation?

The average distance of scores from the mean of µ = 20 is 4 points.


Which symbol below identifies the sample variance?

s^2


Compute the interquartile range for the following scores that represent a continuous variable: 1, 2, 2, 2 4, 5, 10, 12

3


A researcher is interested in examining whether self-affirming oneself leads to decreased anxiety in response to threatening information. The researcher conducts a research study in which an individual is self-affirmed prior to receiving threatening information. Researchers know that on average individuals tend to have anxiety levels of µ = 10 with a standard deviation of σ = 2 when presented with this threatening information. If a self-affirmed individual has an anxiety level of X = 5 following the threatening message, it is reasonable to conclude that self-affirmation ______.

Does have an influence on anxiety levels following threatening information.


A population of scores has µ = 44. In this population, a score of X = 40 corresponds to z = -1.00. What is the population standard deviation?

4


For a population with a standard deviation of σ = 5, what is the z-score corresponding to a score that is 9 points above the mean?

z = +1.80


For a population with µ = 84 and σ = 4, what is the z-score corresponding to X = 80?

-1.00


For a sample, a score that is 12 points above the mean has a z-score of z = 1.20. What is the sample standard deviation?

s = 10


Last week Tim obtained a score of X = 54 on a math exam with µ = 60 and σ = 8. He also scored X = 49 on an English exam with µ = 55 and σ = 3, and he scored X = 31 on a psychology exam with µ = 37 and σ = 4. For which class should Tim expect the worst grade relative to his peers?

English


For a sample with M = 66, a score of X = 55.5 corresponds to z = -1.50. What is the sample standard deviation?

7


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

= 6 (highest score)


A set of scores ranges from a low of X = 25 to a high of X = 33 and has a mean of 29. Which of the following is the most likely value for the standard deviation for these scores

2 points


A sample of n = 6 scores has SS = 40. If these same scores were a population, then the SS for the population would be _____.

40


A population of N = 5 scores has a mean of µ = 20 and a standard deviation of σ = 4. Which of the following is the correct interpretation of the standard deviation?

The average distance of scores from the mean of µ = 20 is 4 points.


Negative skewed

May represent ceiling effects
All data on right, outliers pulling tail to left
Mean<median


Floor effect

A situation in which a constraint prevents a variable from taking values below a certain point


Ceiling effect

A situation in which a constraint prevents a variable from taking on values above a give number


N or n

Number of scores in a data set
N = population
n = sample


E

Summing of set of values


X and/or Y

Individual measurements or scores obtained for a research participant


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